### The Mizar article:

### Products and Coproducts in Categories

**by****Czeslaw Bylinski**

- Received May 11, 1992
Copyright (c) 1992 Association of Mizar Users

- MML identifier: CAT_3
- [ MML identifier index ]

environ vocabulary CAT_1, FUNCT_1, FINSEQ_4, RELAT_1, FUNCOP_1, FUNCT_4, BOOLE, FUNCT_6, OPPCAT_1, WELLORD1, CAT_3; notation TARSKI, XBOOLE_0, SUBSET_1, RELAT_1, FUNCT_1, FUNCOP_1, FUNCT_4, FUNCT_2, FINSEQ_4, CAT_1, OPPCAT_1; constructors OPPCAT_1, FINSEQ_4, MEMBERED, XBOOLE_0; clusters RELSET_1, MEMBERED, ZFMISC_1, XBOOLE_0; requirements SUBSET, BOOLE; definitions CAT_1; theorems TARSKI, FUNCT_2, FUNCOP_1, FUNCT_4, CAT_1, OPPCAT_1, FINSEQ_4, RELSET_1, PARTFUN1; schemes FUNCT_2, COMPTS_1; begin :: Indexed families reserve I for set, x,x1,x2,y,z for set, A for non empty set; reserve C,D for Category; reserve a,b,c,d for Object of C; reserve f,g,h,i,j,k,p1,p2,q1,q2,i1,i2,j1,j2 for Morphism of C; definition let I,A; let F be Function of I,A; let x be set; assume A1: x in I; redefine func F/.x equals :Def1: F.x; compatibility proof let a be Element of A; x in dom F by A1,FUNCT_2:def 1; hence a = F/.x iff a = F.x by FINSEQ_4:def 4; end; end; scheme LambdaIdx{I()->set,A()->non empty set,F(set)->Element of A()}: ex F being Function of I(),A() st for x st x in I() holds F/.x = F(x) proof deffunc f(set) = F($1); A1: for x st x in I() holds f(x) in A(); consider IT being Function of I(),A() such that A2: for x st x in I() holds IT.x = f(x) from Lambda1(A1); take IT; let x; assume A3: x in I(); hence F(x) = IT.x by A2 .= IT/.x by A3,Def1; end; theorem Th1: for F1,F2 being Function of I,A st for x st x in I holds F1/.x = F2/.x holds F1 = F2 proof let F1,F2 be Function of I,A such that A1: for x st x in I holds F1/.x = F2/.x; now let x; assume A2: x in I; hence F1.x = F1/.x by Def1 .= F2/.x by A1,A2 .= F2.x by A2,Def1; end; hence thesis by FUNCT_2:18; end; scheme FuncIdx_correctness{I()->set,A()->non empty set,F(set)->Element of A()}: (ex F being Function of I(),A() st for x st x in I() holds F/.x = F(x)) & (for F1,F2 being Function of I(),A() st (for x st x in I() holds F1/.x = F(x)) & (for x st x in I() holds F2/.x = F(x)) holds F1 = F2) proof deffunc f(set) = F($1); thus ex F being Function of I(),A() st for x st x in I() holds F/.x = f(x) from LambdaIdx; let F1,F2 be Function of I(),A() such that A1: for x st x in I() holds F1/.x = F(x) and A2: for x st x in I() holds F2/.x = F(x); now let x; assume A3: x in I(); hence F1/.x = F(x) by A1 .= F2/.x by A2,A3; end; hence thesis by Th1; end; definition let A,I; let a be Element of A; redefine func I --> a -> Function of I,A; coherence by FUNCOP_1:58; end; theorem Th2: for a being Element of A st x in I holds (I --> a)/.x = a proof let a be Element of A; assume A1: x in I; hence a = (I --> a).x by FUNCOP_1:13 .= (I --> a)/.x by A1,Def1; end; canceled 4; theorem Th7: x1 <> x2 implies for y1,y2 being Element of A holds ((x1,x2) --> (y1,y2))/.x1 = y1 & ((x1,x2) --> (y1,y2))/.x2 = y2 proof assume A1: x1 <> x2; let y1,y2 be Element of A; set h = (x1,x2) --> (y1,y2); h.x1 = y1 & h.x2 = y2 & x1 in {x1,x2} & x2 in {x1,x2} by A1,FUNCT_4:66, TARSKI:def 2; hence thesis by Def1; end; begin :: Indexed families of morphisms definition let C,I; let F be Function of I,the Morphisms of C; canceled; func doms F -> Function of I, the Objects of C means :Def3: for x st x in I holds it/.x = dom(F/.x); correctness proof set A = the Objects of C; deffunc F(set) = dom(F/.$1); thus (ex F being Function of I,A st for x st x in I holds F/.x = F(x)) & (for F1,F2 being Function of I,A st (for x st x in I holds F1/.x = F(x)) & (for x st x in I holds F2/.x = F(x)) holds F1 = F2) from FuncIdx_correctness; end; func cods F -> Function of I, the Objects of C means :Def4: for x st x in I holds it/.x = cod(F/.x); correctness proof set A = the Objects of C; deffunc F(set) = cod(F/.$1); thus (ex F being Function of I,A st for x st x in I holds F/.x = F(x)) & (for F1,F2 being Function of I,A st (for x st x in I holds F1/.x = F(x)) & (for x st x in I holds F2/.x = F(x)) holds F1 = F2) from FuncIdx_correctness; end; end; theorem Th8: doms(I-->f) = I-->(dom f) proof set F = I-->f, F' = I-->(dom f); now let x; assume A1: x in I; then F/.x = f & F'/.x = dom f by Th2; hence (doms F)/.x = F'/.x by A1,Def3; end; hence thesis by Th1; end; theorem Th9: cods(I-->f) = I-->(cod f) proof set F = I-->f, F' = I-->(cod f); now let x; assume A1: x in I; then F/.x = f & F'/.x = cod f by Th2; hence (cods F)/.x = F'/.x by A1,Def4; end; hence thesis by Th1; end; theorem Th10: doms((x1,x2)-->(p1,p2)) = (x1,x2)-->(dom p1,dom p2) proof set F = (x1,x2)-->(p1,p2), f = {x1} --> p1, g = {x2} --> p2, F' = (x1,x2)-->(dom p1,dom p2), f' = {x1}-->(dom p1), g' = {x2}-->(dom p2); A1: dom f = {x1} & dom f' = {x1} & dom g = {x2} & dom g' = {x2} by FUNCOP_1:19; A2: F = f +* g & F' = f' +* g' by FUNCT_4:def 4; now let x; assume A3: x in {x1,x2}; then A4: x in dom F by FUNCT_4:65; now per cases by A2,A4,FUNCT_4:13; case A5: x in dom f & not x in dom g; then F.x = f.x & F'.x = f'.x by A1,A2,FUNCT_4:12; then F.x = p1 & F'.x = dom p1 by A1,A5,FUNCOP_1:13; hence F/.x = p1 & F'/.x = dom p1 by A3,Def1; case A6: x in dom g; then F.x = g.x & F'.x = g'.x by A1,A2,FUNCT_4:14; then F.x = p2 & F'.x = dom p2 by A1,A6,FUNCOP_1:13; hence F/.x = p2 & F'/.x = dom p2 by A3,Def1; end; hence (doms F)/.x = F'/.x by A3,Def3; end; hence thesis by Th1; end; theorem Th11: cods((x1,x2)-->(p1,p2)) = (x1,x2)-->(cod p1,cod p2) proof set F = (x1,x2)-->(p1,p2), f = {x1} --> p1, g = {x2} --> p2, F' = (x1,x2)-->(cod p1,cod p2), f' = {x1}-->(cod p1), g' = {x2}-->(cod p2); A1: dom f = {x1} & dom f' = {x1} & dom g = {x2} & dom g' = {x2} by FUNCOP_1:19; A2: F = f +* g & F' = f' +* g' by FUNCT_4:def 4; now let x; assume A3: x in {x1,x2}; then A4: x in dom F by FUNCT_4:65; now per cases by A2,A4,FUNCT_4:13; case A5: x in dom f & not x in dom g; then F.x = f.x & F'.x = f'.x by A1,A2,FUNCT_4:12; then F.x = p1 & F'.x = cod p1 by A1,A5,FUNCOP_1:13; hence F/.x = p1 & F'/.x = cod p1 by A3,Def1; case A6: x in dom g; then F.x = g.x & F'.x = g'.x by A1,A2,FUNCT_4:14; then F.x = p2 & F'.x = cod p2 by A1,A6,FUNCOP_1:13; hence F/.x = p2 & F'/.x = cod p2 by A3,Def1; end; hence (cods F)/.x = F'/.x by A3,Def4; end; hence thesis by Th1; end; definition let C,I; let F be Function of I,the Morphisms of C; func F opp -> Function of I,the Morphisms of C opp means :Def5: for x st x in I holds it/.x = (F/.x) opp; correctness proof set A = the Morphisms of C opp; deffunc F(set) = (F/.$1) opp; thus (ex F being Function of I,A st for x st x in I holds F/.x = F(x)) & (for F1,F2 being Function of I,A st (for x st x in I holds F1/.x = F(x)) & (for x st x in I holds F2/.x = F(x)) holds F1 = F2) from FuncIdx_correctness; end; end; theorem (I --> f) opp = I --> f opp proof set F = I --> f, F' = I --> f opp; now let x; assume A1: x in I; then F/.x = f & F'/.x = f opp by Th2; hence F opp/.x = F'/.x by A1,Def5; end; hence thesis by Th1; end; theorem x1 <> x2 implies ((x1,x2)-->(p1,p2)) opp = (x1,x2)-->(p1 opp,p2 opp) proof set F = (x1,x2)-->(p1,p2), F' = (x1,x2)-->(p1 opp,p2 opp); assume A1: x1 <> x2; now let x; assume A2: x in {x1,x2}; then x = x1 or x = x2 by TARSKI:def 2; then F/.x = p1 & F'/.x = p1 opp or F/.x = p2 & F'/.x = p2 opp by A1,Th7; hence F opp/.x = F'/.x by A2,Def5; end; hence thesis by Th1; end; theorem for F being Function of I,the Morphisms of C holds F opp opp = F proof let F be Function of I,the Morphisms of C; now C opp = CatStr (#the Objects of C, the Morphisms of C, the Cod of C, the Dom of C, ~(the Comp of C), the Id of C#) & C opp opp = CatStr (#the Objects of C opp, the Morphisms of C opp, the Cod of C opp, the Dom of C opp, ~(the Comp of C opp), the Id of C opp#) by OPPCAT_1:def 1; hence the Morphisms of C = the Morphisms of C opp opp; let x; assume A1: x in I; hence (F opp opp)/.x = ((F opp)/.x) opp by Def5 .= (F/.x) opp opp by A1,Def5 .= F/.x by OPPCAT_1:6; end; hence thesis by Th1; end; definition let C,I; let F be Function of I,the Morphisms of C opp; func opp F -> Function of I,the Morphisms of C means :Def6: for x st x in I holds it/.x = opp (F/.x); correctness proof set A = the Morphisms of C; deffunc F(set) = opp (F/.$1); thus (ex F being Function of I,A st for x st x in I holds F/.x = F(x)) & (for F1,F2 being Function of I,A st (for x st x in I holds F1/.x = F(x)) & (for x st x in I holds F2/.x = F(x)) holds F1 = F2) from FuncIdx_correctness; end; end; theorem for f being Morphism of C opp holds opp(I --> f) = I --> (opp f) proof let f be Morphism of C opp; set F = I --> f, F' = I --> (opp f); now let x; assume A1: x in I; then F/.x = f & F'/.x = opp f by Th2; hence opp F/.x = F'/.x by A1,Def6; end; hence thesis by Th1; end; theorem x1 <> x2 implies for p1,p2 being Morphism of C opp holds opp ((x1,x2)-->(p1,p2)) = (x1,x2)-->(opp p1,opp p2) proof assume A1: x1 <> x2; let p1,p2 be Morphism of C opp; set F = (x1,x2)-->(p1,p2), F' = (x1,x2)-->(opp p1,opp p2); now let x; assume A2: x in {x1,x2}; then x = x1 or x = x2 by TARSKI:def 2; then F/.x = p1 & F'/.x = opp p1 or F/.x = p2 & F'/.x = opp p2 by A1,Th7; hence opp F/.x = F'/.x by A2,Def6; end; hence thesis by Th1; end; theorem for F being Function of I,the Morphisms of C holds opp(F opp) = F proof let F be Function of I,the Morphisms of C; now let x; assume A1: x in I; hence opp(F opp)/.x = opp((F opp)/.x) by Def6 .= opp((F/.x) opp) by A1,Def5 .= F/.x by OPPCAT_1:7; end; hence thesis by Th1; end; definition let C,I; let F be Function of I,the Morphisms of C, f; func F*f -> Function of I,the Morphisms of C means :Def7: for x st x in I holds it/.x = (F/.x)*f; correctness proof set A = the Morphisms of C; deffunc F(set) = (F/.$1)*f; thus (ex F being Function of I,A st for x st x in I holds F/.x = F(x)) & (for F1,F2 being Function of I,A st (for x st x in I holds F1/.x = F(x)) & (for x st x in I holds F2/.x = F(x)) holds F1 = F2) from FuncIdx_correctness; end; func f*F -> Function of I,the Morphisms of C means :Def8: for x st x in I holds it/.x = f*(F/.x); correctness proof set A = the Morphisms of C; deffunc F(set) = f*(F/.$1); thus (ex F being Function of I,A st for x st x in I holds F/.x = F(x)) & (for F1,F2 being Function of I,A st (for x st x in I holds F1/.x = F(x)) & (for x st x in I holds F2/.x = F(x)) holds F1 = F2) from FuncIdx_correctness; end; end; theorem Th18: x1 <> x2 implies ((x1,x2)-->(p1,p2))*f = (x1,x2)-->(p1*f,p2*f) proof assume A1: x1 <> x2; set F = (x1,x2)-->(p1,p2), F' = (x1,x2)-->(p1*f,p2*f); now let x; assume A2: x in {x1,x2}; then x = x1 or x = x2 by TARSKI:def 2; then F/.x = p1 & F'/.x = p1*f or F/.x = p2 & F'/.x = p2*f by A1,Th7; hence (F*f)/.x = F'/.x by A2,Def7; end; hence thesis by Th1; end; theorem Th19: x1 <> x2 implies f*((x1,x2)-->(p1,p2)) = (x1,x2)-->(f*p1,f*p2) proof assume A1: x1 <> x2; set F = (x1,x2)-->(p1,p2), F' = (x1,x2)-->(f*p1,f*p2); now let x; assume A2: x in {x1,x2}; then x = x1 or x = x2 by TARSKI:def 2; then F/.x = p1 & F'/.x = f*p1 or F/.x = p2 & F'/.x = f*p2 by A1,Th7; hence (f*F)/.x = F'/.x by A2,Def8; end; hence thesis by Th1; end; theorem Th20: for F being Function of I,the Morphisms of C st doms F = I-->cod f holds doms(F*f) = I-->dom f & cods(F*f) = cods F proof let F be Function of I,the Morphisms of C such that A1: doms F = I-->cod f; now let x; assume A2: x in I; then A3: dom(F/.x) = (I-->cod f)/.x by A1,Def3 .= cod f by A2,Th2; thus (doms(F*f))/.x = dom((F*f)/.x) by A2,Def3 .= dom((F/.x)*f) by A2,Def7 .= dom f by A3,CAT_1:42 .= (I--> dom f)/.x by A2,Th2; end; hence doms(F*f) = I --> dom f by Th1; now let x; assume A4: x in I; then A5: dom(F/.x) = (I-->cod f)/.x by A1,Def3 .= cod f by A4,Th2; thus (cods F)/.x = cod(F/.x) by A4,Def4 .= cod((F/.x)*f) by A5,CAT_1:42 .= cod((F*f)/.x) by A4,Def7 .= (cods(F*f))/.x by A4,Def4; end; hence cods(F*f) = cods F by Th1; end; theorem Th21: for F being Function of I,the Morphisms of C st cods F = I-->dom f holds doms(f*F) = doms F & cods(f*F) = I-->cod f proof let F be Function of I,the Morphisms of C such that A1: cods F = I-->dom f; now let x; assume A2: x in I; then A3: cod(F/.x) = (I-->dom f)/.x by A1,Def4 .= dom f by A2,Th2; thus (doms F)/.x = dom(F/.x) by A2,Def3 .= dom(f*(F/.x)) by A3,CAT_1:42 .= dom((f*F)/.x) by A2,Def8 .= (doms(f*F))/.x by A2,Def3; end; hence doms(f*F) = doms F by Th1; now let x; assume A4: x in I; then A5: cod(F/.x) = (I-->dom f)/.x by A1,Def4 .= dom f by A4,Th2; thus (cods(f*F))/.x = cod((f*F)/.x) by A4,Def4 .= cod(f*(F/.x)) by A4,Def8 .= cod f by A5,CAT_1:42 .= (I--> cod f)/.x by A4,Th2; end; hence cods(f*F) = I-->cod f by Th1; end; definition let C,I; let F,G be Function of I,the Morphisms of C; func F"*"G -> Function of I,the Morphisms of C means :Def9: for x st x in I holds it/.x = (F/.x)*(G/.x); correctness proof set A = the Morphisms of C; deffunc F(set) = (F/.$1)*(G/.$1); thus (ex F being Function of I,A st for x st x in I holds F/.x = F(x)) & (for F1,F2 being Function of I,A st (for x st x in I holds F1/.x = F(x)) & (for x st x in I holds F2/.x = F(x)) holds F1 = F2) from FuncIdx_correctness; end; end; theorem Th22: for F,G being Function of I,the Morphisms of C st doms F = cods G holds doms(F"*"G) = doms G & cods(F"*"G) = cods F proof let F,G be Function of I,the Morphisms of C such that A1: doms F = cods G; now let x; assume A2: x in I; then A3: cod(G/.x) = (doms F)/.x by A1,Def4 .= dom(F/.x) by A2,Def3; thus (doms(F"*"G))/.x = dom((F"*"G)/.x) by A2,Def3 .= dom((F/.x)*(G/.x)) by A2,Def9 .= dom(G/.x) by A3,CAT_1:42 .= (doms G)/.x by A2,Def3; end; hence doms(F"*"G) = doms G by Th1; now let x; assume A4: x in I; then A5: cod(G/.x) = (doms F)/.x by A1,Def4 .= dom(F/.x) by A4,Def3; thus (cods(F"*"G))/.x = cod((F"*"G)/.x) by A4,Def4 .= cod((F/.x)*(G/.x)) by A4,Def9 .= cod(F/.x) by A5,CAT_1:42 .= (cods F)/.x by A4,Def4; end; hence cods(F"*"G) = cods F by Th1; end; theorem x1 <> x2 implies ((x1,x2)-->(p1,p2))"*"((x1,x2)-->(q1,q2)) = (x1,x2)-->(p1*q1,p2*q2) proof assume A1: x1 <> x2; set F1 = (x1,x2)-->(p1,p2), F2 = (x1,x2)-->(q1,q2), G = (x1,x2)-->(p1*q1,p2*q2); now let x; assume A2: x in {x1,x2}; then x = x1 or x = x2 by TARSKI:def 2; then F1/.x = p1 & F2/.x = q1 & G/.x = p1*q1 or F1/.x = p2 & F2/.x = q2 & G/.x = p2*q2 by A1,Th7; hence (F1"*"F2)/.x = G/.x by A2,Def9; end; hence thesis by Th1; end; theorem for F being Function of I,the Morphisms of C holds F*f = F"*"(I-->f) proof let F be Function of I,the Morphisms of C; now let x; assume A1: x in I; hence (F*f)/.x = (F/.x)*f by Def7 .= (F/.x)*((I-->f)/.x) by A1,Th2 .= (F"*"(I-->f))/.x by A1,Def9; end; hence thesis by Th1; end; theorem for F being Function of I,the Morphisms of C holds f*F = (I-->f)"*"F proof let F be Function of I,the Morphisms of C; now let x; assume A1: x in I; hence (f*F)/.x = f*(F/.x) by Def8 .= ((I-->f)/.x)*(F/.x) by A1,Th2 .= ((I-->f)"*"F)/.x by A1,Def9; end; hence thesis by Th1; end; begin :: Retractions and Coretractions definition let C; let IT be Morphism of C; attr IT is retraction means :Def10: ex g st cod g = dom IT & IT*g = id(cod IT); attr IT is coretraction means :Def11: ex g st dom g = cod IT & g*IT = id(dom IT); end; theorem Th26: f is retraction implies f is epi proof given g such that A1: cod g = dom f and A2: f*g = id(cod f); let p1,p2 such that A3: dom p1 = cod f and A4: dom p2 = cod f and cod p1 = cod p2 and A5: p1*f = p2*f; thus p1 = p1*(f*g) by A2,A3,CAT_1:47 .= (p2*f)*g by A1,A3,A5,CAT_1:43 .= p2*(f*g) by A1,A4,CAT_1:43 .= p2 by A2,A4,CAT_1:47; end; theorem f is coretraction implies f is monic proof given g such that A1: dom g = cod f and A2: g*f = id(dom f); let p1,p2 such that dom p1 = dom p2 and A3: cod p1 = dom f and A4: cod p2 = dom f and A5: f*p1 = f*p2; thus p1 = g*f*p1 by A2,A3,CAT_1:46 .= g*(f*p2) by A1,A3,A5,CAT_1:43 .= g*f*p2 by A1,A4,CAT_1:43 .= p2 by A2,A4,CAT_1:46; end; theorem f is retraction & g is retraction & dom g = cod f implies g*f is retraction proof given i such that A1: cod i = dom f and A2: f*i = id(cod f); given j such that A3: cod j = dom g and A4: g*j = id(cod g); assume A5: dom g = cod f; take i*j; A6: dom i = dom(f*i) by A1,CAT_1:42 .= cod f by A2,CAT_1:44; then A7: cod(i*j) = dom f by A1,A3,A5,CAT_1:42; hence cod(i*j) = dom(g*f) by A5,CAT_1:42; thus g*f*(i*j) = g*(f*(i*j)) by A5,A7,CAT_1:43 .= g*(f*i*j) by A1,A3,A5,A6,CAT_1:43 .= id(cod g) by A2,A3,A4,A5,CAT_1:46 .= id(cod(g*f)) by A5,CAT_1:42; end; theorem f is coretraction & g is coretraction & dom g = cod f implies g*f is coretraction proof given i such that A1: dom i = cod f and A2: i*f = id(dom f); given j such that A3: dom j = cod g and A4: j*g = id(dom g); assume A5: dom g = cod f; take i*j; A6: cod j = cod(j*g) by A3,CAT_1:42 .= dom g by A4,CAT_1:44; A7: cod g = cod(g*f) by A5,CAT_1:42; hence dom(i*j) = cod(g*f) by A1,A3,A5,A6,CAT_1:42; thus i*j*(g*f) = i*(j*(g*f)) by A1,A3,A5,A6,A7,CAT_1:43 .= i*(j*g*f) by A3,A5,CAT_1:43 .= id(dom f) by A2,A4,A5,CAT_1:46 .= id(dom(g*f)) by A5,CAT_1:42; end; theorem dom g = cod f & g*f is retraction implies g is retraction proof assume A1: dom g = cod f; given i such that A2: cod i = dom(g*f) and A3: (g*f)*i = id(cod(g*f)); take f*i; A4: dom f = dom(g*f) by A1,CAT_1:42; hence cod(f*i) = dom g by A1,A2,CAT_1:42; thus g*(f*i) = id(cod(g*f)) by A1,A2,A3,A4,CAT_1:43 .= id(cod g) by A1,CAT_1:42; end; theorem dom g = cod f & g*f is coretraction implies f is coretraction proof assume A1: dom g = cod f; given i such that A2: dom i = cod(g*f) and A3: i*(g*f) = id(dom(g*f)); take i*g; A4: cod g = cod(g*f) by A1,CAT_1:42; hence dom(i*g) = cod f by A1,A2,CAT_1:42; thus (i*g)*f = id(dom(g*f)) by A1,A2,A3,A4,CAT_1:43 .= id(dom f) by A1,CAT_1:42; end; theorem f is retraction & f is monic implies f is invertible proof given i such that A1: cod i = dom f and A2: f*i = id(cod f); assume A3: f is monic; take i; thus A4: dom i = dom(f*i) by A1,CAT_1:42 .= cod f by A2,CAT_1:44; thus cod i = dom f by A1; thus f*i = id cod f by A2; now thus dom(i*f) = dom f by A4,CAT_1:42 .= dom(id(dom f)) by CAT_1:44; thus cod(i*f) = dom f by A1,A4,CAT_1:42; thus cod(id(dom f)) = dom f by CAT_1:44; thus f*(i*f) = id(cod f)*f by A1,A2,A4,CAT_1:43 .= f by CAT_1:46 .= f * id(dom f) by CAT_1:47; end; hence i*f = id(dom f) by A3,CAT_1:def 10; end; theorem Th33: f is coretraction & f is epi implies f is invertible proof given i such that A1: dom i = cod f and A2: i*f = id(dom f); assume A3: f is epi; take i; thus dom i = cod f by A1; thus A4: cod i = cod(i*f) by A1,CAT_1:42 .= dom f by A2,CAT_1:44; now thus dom(f*i) = cod f by A1,A4,CAT_1:42; thus dom(id(cod f)) = cod f by CAT_1:44; thus cod(f*i) = cod f by A4,CAT_1:42 .= cod(id(cod f)) by CAT_1:44; thus f*i*f = f * id(dom f) by A1,A2,A4,CAT_1:43 .= f by CAT_1:47 .= id(cod f)*f by CAT_1:46; end; hence f*i = id cod f by A3,CAT_1:def 11; thus i*f = id(dom f) by A2; end; theorem f is invertible iff f is retraction & f is coretraction proof thus f is invertible implies f is retraction & f is coretraction proof assume ex g st dom g = cod f & cod g = dom f & f*g = id cod f & g*f = id dom f; hence thesis by Def10,Def11; end; assume f is retraction; then f is epi by Th26; hence thesis by Th33; end; theorem for T being Functor of C,D st f is retraction holds T.f is retraction proof let T be Functor of C,D; given i such that A1: cod i = dom f and A2: f*i = id(cod f); take T.i; thus cod(T.i) = T.(dom f) by A1,CAT_1:109 .= dom(T.f) by CAT_1:109; thus (T.f)*(T.i) = T.(id(cod f)) by A1,A2,CAT_1:99 .= id(cod(T.f)) by CAT_1:98; end; theorem for T being Functor of C,D st f is coretraction holds T.f is coretraction proof let T be Functor of C,D; given i such that A1: dom i = cod f and A2: i*f = id(dom f); take T.i; thus dom(T.i) = T.(cod f) by A1,CAT_1:109 .= cod(T.f) by CAT_1:109; thus (T.i)*(T.f) = T.(id(dom f)) by A1,A2,CAT_1:99 .= id(dom(T.f)) by CAT_1:98; end; theorem f is retraction iff f opp is coretraction proof thus f is retraction implies f opp is coretraction proof given i such that A1: cod i = dom f and A2: f*i = id(cod f); take i opp; thus dom(i opp) = dom f by A1,OPPCAT_1:9 .= cod(f opp) by OPPCAT_1:9; thus (i opp)*(f opp) = (id(cod f)) opp by A1,A2,OPPCAT_1:17 .= id((cod f)opp) by OPPCAT_1:21 .= id(dom(f opp)) by OPPCAT_1:11; end; given i being Morphism of C opp such that A3: dom i = cod(f opp) and A4: i*(f opp) = id(dom(f opp)); take opp i; thus A5: cod(opp i) = cod(f opp) by A3,OPPCAT_1:10 .= dom f by OPPCAT_1:9; thus f*(opp i) = (f*(opp i)) opp by OPPCAT_1:def 4 .= ((opp i)opp)*(f opp) by A5,OPPCAT_1:17 .= id(dom(f opp)) by A4,OPPCAT_1:8 .= id((cod f)opp) by OPPCAT_1:11 .= (id(cod f))opp by OPPCAT_1:21 .= id(cod f) by OPPCAT_1:def 4; end; theorem f is coretraction iff f opp is retraction proof thus f is coretraction implies f opp is retraction proof given i such that A1: dom i = cod f and A2: i*f = id(dom f); take i opp; thus cod(i opp) = cod f by A1,OPPCAT_1:9 .= dom(f opp) by OPPCAT_1:9; thus (f opp)*(i opp) = (id(dom f))opp by A1,A2,OPPCAT_1:17 .= id((dom f)opp) by OPPCAT_1:21 .= id(cod(f opp)) by OPPCAT_1:11; end; given i being Morphism of C opp such that A3: cod i = dom(f opp) and A4: (f opp)*i = id(cod(f opp)); take opp i; thus A5: dom(opp i) = dom(f opp) by A3,OPPCAT_1:10 .= cod f by OPPCAT_1:9; thus (opp i)*f = ((opp i)*f) opp by OPPCAT_1:def 4 .= (f opp)*((opp i)opp) by A5,OPPCAT_1:17 .= id(cod(f opp)) by A4,OPPCAT_1:8 .= id((dom f)opp) by OPPCAT_1:11 .= (id(dom f))opp by OPPCAT_1:21 .= id(dom f) by OPPCAT_1:def 4; end; begin :: Morphisms determined by a terminal Object definition let C,a,b; assume A1: b is terminal; func term(a,b) -> Morphism of a,b means not contradiction; existence; uniqueness proof let f1,f2 be Morphism of a,b; consider f being Morphism of a,b such that A2: for g being Morphism of a,b holds f = g by A1,CAT_1:def 15; thus f1 = f by A2 .= f2 by A2; end; end; theorem Th39: b is terminal implies dom(term(a,b)) = a & cod(term(a,b)) = b proof assume b is terminal; then Hom(a,b) <> {} by CAT_1:def 15; hence thesis by CAT_1:23; end; theorem Th40: b is terminal & dom f = a & cod f = b implies term(a,b) = f proof assume that A1: b is terminal and A2: dom f = a & cod f = b; consider h being Morphism of a,b such that A3: for g being Morphism of a,b holds h = g by A1,CAT_1:def 15; f is Morphism of a,b by A2,CAT_1:22; hence f = h by A3 .= term(a,b) by A3; end; theorem for f being Morphism of a,b st b is terminal holds term(a,b) = f proof let f be Morphism of a,b; assume A1: b is terminal; then Hom(a,b) <> {} by CAT_1:def 15; then dom f = a & cod f = b by CAT_1:23; hence thesis by A1,Th40; end; begin :: Morphisms determined by an iniatial object definition let C,a,b; assume A1: a is initial; func init(a,b) -> Morphism of a,b means not contradiction; existence; uniqueness proof let f1,f2 be Morphism of a,b; consider f being Morphism of a,b such that A2: for g being Morphism of a,b holds f = g by A1,CAT_1:def 16; thus f1 = f by A2 .= f2 by A2; end; end; theorem Th42: a is initial implies dom(init(a,b)) = a & cod(init(a,b)) = b proof assume a is initial; then Hom(a,b) <> {} by CAT_1:def 16; hence dom(init(a,b)) = a & cod(init(a,b)) = b by CAT_1:23; end; theorem Th43: a is initial & dom f = a & cod f = b implies init(a,b) = f proof assume that A1: a is initial and A2: dom f = a & cod f = b; consider h being Morphism of a,b such that A3: for g being Morphism of a,b holds h = g by A1,CAT_1:def 16; f is Morphism of a,b by A2,CAT_1:22; hence f = h by A3 .= init(a,b) by A3; end; theorem for f being Morphism of a,b st a is initial holds init(a,b) = f proof let f be Morphism of a,b; assume A1: a is initial; then Hom(a,b) <> {} by CAT_1:def 16; then dom f = a & cod f = b by CAT_1:23; hence thesis by A1,Th43; end; begin :: Products definition let C,a,I; mode Projections_family of a,I -> Function of I,the Morphisms of C means :Def14: doms it = I --> a; existence proof take F = I-->id a; doms F = I --> (dom id a) by Th8; hence thesis by CAT_1:44; end; end; theorem Th45: for F being Projections_family of a,I st x in I holds dom(F/.x) = a proof let F be Projections_family of a,I such that A1: x in I; (doms F)/.x = (I --> a)/.x by Def14; hence dom(F/.x) = (I --> a)/.x by A1,Def3 .= a by A1,Th2; end; theorem Th46: for F being Function of {},the Morphisms of C holds F is Projections_family of a,{} proof let F be Function of {},the Morphisms of C; thus {} --> a = {} by FUNCT_4:3 .= doms F by PARTFUN1:57; end; theorem Th47: dom f = a implies {y} --> f is Projections_family of a,{y} proof set F = {y} --> f; assume A1: dom f = a; now let x; assume A2: x in {y}; hence (doms F)/.x = dom(F/.x) by Def3 .= a by A1,A2,Th2 .= ({y} --> a)/.x by A2,Th2; end; hence doms F = {y} --> a by Th1; end; theorem Th48: dom p1 = a & dom p2 = a implies (x1,x2)-->(p1,p2) is Projections_family of a,{x1,x2} proof assume A1: dom p1 = a & dom p2 = a; doms ((x1,x2)-->(p1,p2)) = (x1,x2) --> (dom p1,dom p2) by Th10 .= {x1,x2} --> a by A1,FUNCT_4:68; hence thesis by Def14; end; canceled; theorem Th50: for F being Projections_family of a,I st cod f = a holds F*f is Projections_family of dom f,I proof let F be Projections_family of a,I; assume cod f = a; then doms F = I-->cod f by Def14; hence doms(F*f) = I --> dom f by Th20; end; theorem for F being Function of I,the Morphisms of C, G being Projections_family of a,I st doms F = cods G holds F"*"G is Projections_family of a,I proof let F be Function of I,the Morphisms of C; let G be Projections_family of a,I; assume doms F = cods G; then doms(F"*"G) = doms G by Th22; hence doms(F"*"G) = I --> a by Def14; end; theorem for F being Projections_family of cod f,I holds (f opp)*(F opp) = (F*f) opp proof let F be Projections_family of cod f, I; now let x; assume A1: x in I; then A2: dom(F/.x) = (doms F)/.x by Def3 .= (I --> cod f)/.x by Def14 .= cod f by A1,Th2; thus ((f opp)*(F opp))/.x = (f opp)*((F opp)/.x) by A1,Def8 .= (f opp)*((F/.x)opp) by A1,Def5 .= ((F/.x)*f)opp by A2,OPPCAT_1:17 .= ((F*f)/.x)opp by A1,Def7 .= ((F*f) opp)/.x by A1,Def5; end; hence thesis by Th1; end; definition let C,a,I; let F be Function of I,the Morphisms of C; pred a is_a_product_wrt F means :Def15: F is Projections_family of a,I & for b for F' being Projections_family of b,I st cods F = cods F' ex h st h in Hom(b,a) & for k st k in Hom(b,a) holds (for x st x in I holds (F/.x)*k = F'/.x) iff h = k; end; theorem Th53: for F being Projections_family of c,I, F' being Projections_family of d,I st c is_a_product_wrt F & d is_a_product_wrt F' & cods F = cods F' holds c,d are_isomorphic proof let F be Projections_family of c,I, F' be Projections_family of d,I such that A1: c is_a_product_wrt F and A2: d is_a_product_wrt F' and A3: cods F = cods F'; consider f such that A4: f in Hom(d,c) and A5: for k st k in Hom(d,c) holds (for x st x in I holds (F/.x)*k = F'/.x ) iff f = k by A1,A3,Def15; consider g such that A6: g in Hom(c,d) and A7: for k st k in Hom(c,d) holds (for x st x in I holds (F'/.x)*k = F/.x ) iff g = k by A2,A3,Def15; thus Hom(c,d) <> {} by A6; reconsider g as Morphism of c,d by A6,CAT_1:def 7; take g,f; A8: dom f = d & cod g = d & cod f = c & dom g = c by A4,A6,CAT_1:18; hence dom f = cod g & cod f = dom g; cods F' = cods F'; then consider gf being Morphism of C such that gf in Hom(d,d) and A9: for k st k in Hom(d,d) holds (for x st x in I holds (F'/.x)*k = F'/.x ) iff gf = k by A2,Def15; A10: now set k = id d; thus k in Hom(d,d) by CAT_1:55; let x; assume x in I; then dom(F'/.x) = d by Th45; hence (F'/.x)*k = F'/.x by CAT_1:47; end; now dom(g*f) = d & cod(g*f) = d by A8,CAT_1:42; hence g*f in Hom(d,d) by CAT_1:18; let x; assume A11: x in I; then dom(F'/.x) = d by Th45; hence (F'/.x)*(g*f) = (F'/.x)*g*f by A8,CAT_1:43 .= (F/.x)*f by A6,A7,A11 .= F'/.x by A4,A5,A11; end; hence g*f = gf by A9 .= id cod g by A8,A9,A10; A12: now set k = id c; thus k in Hom(c,c) by CAT_1:55; let x; assume x in I; then dom(F/.x) = c by Th45; hence (F/.x)*k = F/.x by CAT_1:47; end; cods F = cods F; then consider fg being Morphism of C such that fg in Hom(c,c) and A13: for k st k in Hom(c,c) holds (for x st x in I holds (F/.x)*k = F/.x ) iff fg = k by A1,Def15; now dom(f*g) = c & cod(f*g) = c by A8,CAT_1:42; hence f*g in Hom(c,c) by CAT_1:18; let x; assume A14: x in I; then dom(F/.x) = c by Th45; hence (F/.x)*(f*g) = (F/.x)*f*g by A8,CAT_1:43 .= (F'/.x)*g by A4,A5,A14 .= F/.x by A6,A7,A14; end; hence f*g = fg by A13 .= id dom g by A8,A12,A13; end; theorem Th54: for F being Projections_family of c,I st c is_a_product_wrt F & for x1,x2 st x1 in I & x2 in I holds Hom(cod(F/.x1),cod(F/.x2)) <> {} for x st x in I holds F/.x is retraction proof let F be Projections_family of c,I such that A1: c is_a_product_wrt F and A2: for x1,x2 st x1 in I & x2 in I holds Hom(cod(F/.x1),cod(F/.x2)) <> {}; let x such that A3: x in I; set d = cod(F/.x); defpred P[set,set] means ($1 = x implies $2 = id d) & ($1 <> x implies $2 in Hom(d,cod(F/.$1))); A4: for y st y in I holds ex z st z in the Morphisms of C & P[y,z] proof let y; assume y in I; then A5: Hom(d,cod(F/.y)) <> {} by A2,A3; consider z being Element of Hom(d,cod(F/.y)); A6: z is Morphism of d,cod(F/.y) by A5,CAT_1:21; per cases; suppose A7: y = x; take z = id d; thus z in the Morphisms of C; thus (y = x implies z = id d) & (y <> x implies z in Hom(d,cod(F/.y))) by A7; suppose A8: y <> x; take z; thus z in the Morphisms of C by A6; thus (y = x implies z = id d) & (y <> x implies z in Hom(d,cod(F/.y))) by A5,A8; end; consider F' being Function such that A9: dom F' = I & rng F' c= the Morphisms of C and A10: for y st y in I holds P[y,F'.y] from NonUniqBoundFuncEx(A4); reconsider F' as Function of I,the Morphisms of C by A9,FUNCT_2:def 1, RELSET_1:11; now now let y; assume A11: y in I; then F'.y = F'/.y & F'.x = F'/.x by A3,Def1; then (y = x implies F'/.y = id d) & (y <> x implies F'/.y in Hom(d,cod(F/.y))) by A10,A11; then dom(F'/.y) = d by CAT_1:18,44; hence (doms F')/.y = d by A11,Def3 .= (I-->d)/.y by A11,Th2; end; hence doms F' = I --> d by Th1; hence F' is Projections_family of d,I by Def14; now let y; assume A12: y in I; then F'.y = F'/.y & F'.x = F'/.x by A3,Def1; then (y = x implies F'/.y = id d) & (y <> x implies F'/.y in Hom(d,cod(F/.y))) by A10,A12; then cod(F'/.y) = cod(F/.y) by CAT_1:18,44; hence (cods F')/.y = cod(F/.y) by A12,Def4 .= (cods F)/.y by A12,Def4; end; hence cods F = cods F' by Th1; end; then consider i such that A13: i in Hom(d,c) and A14: for k st k in Hom(d,c) holds (for y st y in I holds (F/.y)*k = F'/.y) iff i = k by A1,Def15; take i; thus cod i = c by A13,CAT_1:18 .= dom(F/.x) by A3,Th45; thus (F/.x)*i = F'/.x by A3,A13,A14 .= F'.x by A3,Def1 .= id d by A3,A10; end; theorem Th55: for F being Function of {},the Morphisms of C holds a is_a_product_wrt F iff a is terminal proof let F be Function of {},the Morphisms of C; thus a is_a_product_wrt F implies a is terminal proof assume A1: a is_a_product_wrt F; let b; consider F' being Projections_family of b,{}; cods F = {} & cods F' = {} by PARTFUN1:57; then consider h such that A2: h in Hom(b,a) and A3: for k st k in Hom(b,a) holds (for x st x in {} holds (F/.x)*k = F'/.x ) iff h = k by A1,Def15; thus Hom(b,a)<>{} by A2; reconsider f = h as Morphism of b,a by A2,CAT_1:def 7; take f; let g be Morphism of b,a; g in Hom(b,a) & for x st x in {} holds (F/.x)*g = F'/.x by A2,CAT_1:def 7; hence f=g by A3; end; assume A4: a is terminal; thus F is Projections_family of a,{} by Th46; let b; let F' be Projections_family of b,{} such that cods F = cods F'; consider h being Morphism of b,a such that A5: for g being Morphism of b,a holds h = g by A4,CAT_1:def 15; take h; Hom(b,a)<>{} by A4,CAT_1:def 15; hence h in Hom(b,a) by CAT_1:def 7; let k; assume k in Hom(b,a); then k is Morphism of b,a by CAT_1:def 7; hence (for x st x in {} holds (F/.x)*k = F'/.x ) iff h = k by A5; end; theorem Th56: for F being Projections_family of a,I st a is_a_product_wrt F & dom f = b & cod f = a & f is invertible holds b is_a_product_wrt F*f proof let F be Projections_family of a,I such that A1: a is_a_product_wrt F and A2: dom f = b & cod f = a and A3: f is invertible; thus F*f is Projections_family of b,I by A2,Th50; let c; let F' be Projections_family of c,I such that A4: cods(F*f) = cods F'; doms F = I-->cod f by A2,Def14; then cods F = cods F' by A4,Th20; then consider h such that A5: h in Hom(c,a) and A6: for k st k in Hom(c,a) holds (for x st x in I holds (F/.x)*k = F'/.x) iff h = k by A1,Def15; consider g such that A7: dom g = cod f & cod g = dom f and A8: f*g = id cod f and A9: g*f = id dom f by A3,CAT_1:def 12; take gh = g*h; A10: cod h = a & dom h = c by A5,CAT_1:18; then dom(g*h) = c & cod(g*h) = b by A2,A7,CAT_1:42; hence gh in Hom(c,b) by CAT_1:18; let k; assume k in Hom(c,b); then A11: dom k = c & cod k = b by CAT_1:18; thus (for x st x in I holds ((F*f)/.x)*k = F'/.x) implies gh = k proof assume A12: for x st x in I holds ((F*f)/.x)*k = F'/.x; now dom(f*k) = c & cod(f*k) = a by A2,A11,CAT_1:42; hence f*k in Hom(c,a) by CAT_1:18; let x; assume A13: x in I; then dom(F/.x) = a by Th45; hence (F/.x)*(f*k) = (F/.x)*f*k by A2,A11,CAT_1:43 .= ((F*f)/.x)*k by A13,Def7 .= F'/.x by A12,A13; end; then g*(f*k) = g*h by A6; hence gh = (id b)*k by A2,A7,A9,A11,CAT_1:43 .= k by A11,CAT_1:46; end; assume A14: gh = k; let x; assume A15: x in I; then A16: dom(F/.x) = a by Th45; thus ((F*f)/.x)*k = ((F/.x)*f)*k by A15,Def7 .= (F/.x)*(f*(g*h)) by A2,A11,A14,A16,CAT_1:43 .= (F/.x)*((id cod f)*h) by A2,A7,A8,A10,CAT_1:43 .= (F/.x)*h by A2,A10,CAT_1:46 .= F'/.x by A5,A6,A15; end; theorem a is_a_product_wrt {y} --> id a proof set F = {y} --> id a; dom(id a) = a by CAT_1:44; hence F is Projections_family of a,{y} by Th47; let b; let F' be Projections_family of b,{y} such that A1: cods F = cods F'; take h = F'/.y; A2: y in {y} by TARSKI:def 1; now thus dom h = b by A2,Th45; thus cod h = (cods F)/.y by A1,A2,Def4 .= cod(F/.y) by A2,Def4 .= cod(id a) by A2,Th2 .= a by CAT_1:44; end; hence h in Hom(b,a) by CAT_1:18; let k; assume k in Hom(b,a); then A3: cod k = a by CAT_1:18; thus (for x st x in {y} holds (F/.x)*k = F'/.x) implies h = k proof assume A4: for x st x in {y} holds (F/.x)*k = F'/.x; thus k = (id a)*k by A3,CAT_1:46 .= (F/.y)*k by A2,Th2 .= h by A2,A4; end; assume A5: h = k; let x; assume A6: x in {y}; hence F'/.x = k by A5,TARSKI:def 1 .= (id a)*k by A3,CAT_1:46 .= (F/.x)*k by A6,Th2; end; theorem for F being Projections_family of a,I st a is_a_product_wrt F & for x st x in I holds cod(F/.x) is terminal holds a is terminal proof let F be Projections_family of a,I such that A1: a is_a_product_wrt F and A2: for x st x in I holds cod(F/.x) is terminal; now thus I = {} implies a is terminal by A1,Th55; let b; deffunc f(set) = term(b,cod(F/.$1)); consider F' being Function of I, the Morphisms of C such that A3: for x st x in I holds F'/.x = f(x) from LambdaIdx; now let x; assume A4: x in I; then A5: cod(F/.x) is terminal by A2; thus (doms F')/.x = dom(F'/.x) by A4,Def3 .= dom(term(b,cod(F/.x))) by A3,A4 .= b by A5,Th39 .= (I-->b)/.x by A4,Th2; end; then doms F' = I --> b by Th1; then reconsider F' as Projections_family of b,I by Def14; now let x; assume A6: x in I; then A7: cod(F/.x) is terminal by A2; thus (cods F)/.x = cod(F/.x) by A6,Def4 .= cod(term(b,cod(F/.x))) by A7,Th39 .= cod(F'/.x) by A3,A6 .= (cods F')/.x by A6,Def4; end; then cods F = cods F' by Th1; then consider h such that A8: h in Hom(b,a) and A9: for k st k in Hom(b,a) holds (for x st x in I holds (F/.x)*k = F'/.x) iff h = k by A1,Def15; thus Hom(b,a)<>{} by A8; reconsider h as Morphism of b,a by A8,CAT_1:def 7; take h; let g be Morphism of b,a; now thus g in Hom(b,a) by A8,CAT_1:def 7; let x; set c = cod(F/.x); assume A10: x in I; then c is terminal by A2; then consider f being Morphism of b,c such that A11: for f' being Morphism of b,c holds f = f' by CAT_1:def 15; dom g = b & cod g = a & dom(F/.x) = a by A8,A10,Th45,CAT_1:23; then dom((F/.x)*g) = b & cod((F/.x)*g) = c by CAT_1:42; then (F/.x)*g in Hom(b,c) by CAT_1:18; then A12: (F/.x)*g is Morphism of b,c by CAT_1:def 7; F'/.x = term(b,c) by A3,A10; hence F'/.x = f by A11 .= (F/.x)*g by A11,A12; end; hence h = g by A9; end; hence thesis by CAT_1:def 15; end; definition let C,c,p1,p2; pred c is_a_product_wrt p1,p2 means :Def16: dom p1 = c & dom p2 = c & for d,f,g st f in Hom(d,cod p1) & g in Hom(d,cod p2) ex h st h in Hom(d,c) & for k st k in Hom(d,c) holds p1*k = f & p2*k = g iff h = k; end; theorem Th59: x1 <> x2 implies (c is_a_product_wrt p1,p2 iff c is_a_product_wrt (x1,x2)-->(p1,p2)) proof set F = (x1,x2)-->(p1,p2), I = {x1,x2}; assume A1: x1 <> x2; thus c is_a_product_wrt p1,p2 implies c is_a_product_wrt F proof assume A2: c is_a_product_wrt p1,p2; then dom p1 = c & dom p2 = c by Def16; hence F is Projections_family of c,I by Th48; let b; let F' be Projections_family of b,I such that A3: cods F = cods F'; set f = F'/.x1, g = F'/.x2; A4: x1 in I & x2 in I by TARSKI:def 2; then (cods F)/.x1 = cod(F/.x1) & (cods F)/.x2 = cod(F/.x2) by Def4; then cod f = cod(F/.x1) & cod g = cod(F/.x2) by A3,A4,Def4; then cod f = cod p1 & cod g = cod p2 & dom f = b & dom g = b by A1,A4,Th7,Th45; then f in Hom(b,cod p1) & g in Hom(b,cod p2) by CAT_1:18; then consider h such that A5: h in Hom(b,c) and A6: for k st k in Hom(b,c) holds p1*k = f & p2*k = g iff h = k by A2,Def16; take h; thus h in Hom(b,c) by A5; let k such that A7: k in Hom(b,c); thus (for x st x in I holds (F/.x)*k = F'/.x ) implies h = k proof assume for x st x in I holds (F/.x)*k = F'/.x; then (F/.x1)*k = f & (F/.x2)*k = g by A4; then p1*k = f & p2*k = g by A1,Th7; hence h = k by A6,A7; end; assume A8: h = k; let x such that A9: x in I; p1*k = f & p2*k = g by A6,A7,A8; then (F/.x1)*k = f & (F/.x2)*k = g & (x = x1 or x = x2) by A1,A9,Th7,TARSKI:def 2; hence thesis; end; assume A10: c is_a_product_wrt F; then x1 in I & x2 in I & F is Projections_family of c,I by Def15,TARSKI:def 2; then dom(F/.x1) = c & dom(F/.x2) = c by Th45; hence dom p1 = c & dom p2 = c by A1,Th7; let d,f,g such that A11: f in Hom(d,cod p1) and A12: g in Hom(d,cod p2); dom f = d & dom g = d by A11,A12,CAT_1:18; then reconsider F' = (x1,x2) --> (f,g) as Projections_family of d,I by Th48; cods F = (x1,x2)-->(cod p1,cod p2) by Th11 .= (x1,x2)-->(cod f,cod p2) by A11,CAT_1:18 .= (x1,x2)-->(cod f,cod g) by A12,CAT_1:18 .= cods F' by Th11; then consider h such that A13: h in Hom(d,c) and A14: for k st k in Hom(d,c) holds (for x st x in I holds (F/.x)*k = F'/.x ) iff h = k by A10,Def15; take h; thus h in Hom(d,c) by A13; let k such that A15: k in Hom(d,c); thus p1*k = f & p2*k = g implies h = k proof assume A16: p1*k = f & p2*k = g; now let x; assume x in I; then x = x1 or x = x2 by TARSKI:def 2; then F/.x = p1 & F'/.x = f or F/.x = p2 & F'/.x = g by A1,Th7; hence (F/.x)*k = F'/.x by A16; end; hence h = k by A14,A15; end; assume h = k; then x1 in I & x2 in I & for x st x in I holds (F/.x)*k = F'/.x by A14,A15,TARSKI:def 2; then (F/.x1)*k = F'/.x1 & (F/.x2)*k = F'/.x2; then (F/.x1)*k = f & (F/.x2)*k = g by A1,Th7; hence p1*k = f & p2*k = g by A1,Th7; end; theorem Hom(c,a) <> {} & Hom(c,b) <> {} implies for p1 being Morphism of c,a, p2 being Morphism of c,b holds c is_a_product_wrt p1,p2 iff for d st Hom(d,a)<>{} & Hom(d,b)<>{} holds Hom(d,c) <> {} & for f being Morphism of d,a, g being Morphism of d,b ex h being Morphism of d,c st for k being Morphism of d,c holds p1*k = f & p2*k = g iff h = k proof assume A1: Hom(c,a) <> {} & Hom(c,b) <> {}; let p1 be Morphism of c,a, p2 be Morphism of c,b; thus c is_a_product_wrt p1,p2 implies for d st Hom(d,a)<>{} & Hom(d,b)<>{} holds Hom(d,c) <> {} & for f being Morphism of d,a, g being Morphism of d,b ex h being Morphism of d,c st for k being Morphism of d,c holds p1*k = f & p2*k = g iff h = k proof assume that dom p1 = c & dom p2 = c and A2: for d,f,g st f in Hom(d,cod p1) & g in Hom(d,cod p2) ex h st h in Hom(d,c) & for k st k in Hom(d,c) holds p1*k = f & p2*k = g iff h = k; let d such that A3: Hom(d,a)<>{} & Hom(d,b)<>{}; consider f being Morphism of d,a, g being Morphism of d,b; A4: cod p1 = a & cod p2 = b by A1,CAT_1:23; then f in Hom(d,cod p1) & g in Hom(d,cod p2) by A3,CAT_1:def 7; then A5: ex h st h in Hom(d,c) & for k st k in Hom(d,c) holds p1*k = f & p2*k = g iff h = k by A2; hence Hom(d,c) <> {}; let f be Morphism of d,a, g be Morphism of d,b; f in Hom(d,cod p1) & g in Hom(d,cod p2) by A3,A4,CAT_1:def 7; then consider h such that A6: h in Hom(d,c) and A7: for k st k in Hom(d,c) holds p1*k = f & p2*k = g iff h = k by A2; reconsider h as Morphism of d,c by A6,CAT_1:def 7; take h; let k be Morphism of d,c; p1*k = p1*(k qua Morphism of C) & p2*k = p2*(k qua Morphism of C) & k in Hom(d,c) by A1,A5,CAT_1:def 7,def 13; hence thesis by A7; end; assume A8: for d st Hom(d,a)<>{} & Hom(d,b)<>{} holds Hom(d,c) <> {} & for f being Morphism of d,a, g being Morphism of d,b ex h being Morphism of d,c st for k being Morphism of d,c holds p1*k = f & p2*k = g iff h = k; thus dom p1 = c & dom p2 = c by A1,CAT_1:23; let d,f,g such that A9: f in Hom(d,cod p1) and A10: g in Hom(d,cod p2); A11: cod p1 = a & cod p2 = b by A1,CAT_1:23; then f is Morphism of d,a & g is Morphism of d,b & Hom(d,a) <> {} & Hom(d,b) <> {} by A9,A10,CAT_1:def 7; then consider h being Morphism of d,c such that A12: for k being Morphism of d,c holds p1*k = f & p2*k = g iff h = k by A8; reconsider h' = h as Morphism of C; take h'; A13: Hom(d,c) <> {} by A8,A9,A10,A11; hence h' in Hom(d,c) by CAT_1:def 7; let k; assume k in Hom(d,c); then reconsider k' = k as Morphism of d,c by CAT_1:def 7; p1*k = p1*k' & p2*k = p2*k' by A1,A13,CAT_1:def 13; hence thesis by A12; end; theorem c is_a_product_wrt p1,p2 & d is_a_product_wrt q1,q2 & cod p1 = cod q1 & cod p2 = cod q2 implies c,d are_isomorphic proof assume that A1: c is_a_product_wrt p1,p2 and A2: d is_a_product_wrt q1,q2 and A3: cod p1 = cod q1 & cod p2 = cod q2; set I = {0,1}, F = (0,1)-->(p1,p2), F' = (0,1)-->(q1,q2); now dom p1 = c & dom p2 = c & dom q1 = d & dom q2 = d by A1,A2,Def16; hence F is Projections_family of c,I & F' is Projections_family of d,I by Th48; thus c is_a_product_wrt F & d is_a_product_wrt F' by A1,A2,Th59; thus cods F = (0,1)-->(cod q1,cod q2) by A3,Th11 .= cods F' by Th11; end; hence thesis by Th53; end; theorem c is_a_product_wrt p1,p2 & Hom(cod p1,cod p2) <> {} & Hom(cod p2,cod p1) <> {} implies p1 is retraction & p2 is retraction proof assume that A1: c is_a_product_wrt p1,p2 and A2: Hom(cod p1,cod p2) <> {} & Hom(cod p2,cod p1) <> {}; set I = {0,1}, F = (0,1)-->(p1,p2); A3: F/.0 = p1 & F/.1 = p2 by Th7; now dom p1 = c & dom p2 = c by A1,Def16; hence F is Projections_family of c,I by Th48; thus c is_a_product_wrt F by A1,Th59; let x1,x2; assume x1 in I & x2 in I; then (x1 = 0 or x1 = 1) & (x2 = 0 or x2 = 1) by TARSKI:def 2; hence Hom(cod(F/.x1),cod(F/.x2)) <> {} by A2,A3,CAT_1:56; end; then 0 in I & 1 in I & for x st x in I holds F/.x is retraction by Th54,TARSKI:def 2; hence thesis by A3; end; theorem Th63: c is_a_product_wrt p1,p2 & h in Hom(c,c) & p1*h = p1 & p2*h = p2 implies h = id c proof assume that A1: dom p1 = c and A2: dom p2 = c and A3: for d,f,g st f in Hom(d,cod p1) & g in Hom(d,cod p2) ex h st h in Hom(d,c) & for k st k in Hom(d,c) holds p1*k = f & p2*k = g iff h = k and A4: h in Hom(c,c) and A5: p1*h = p1 & p2*h = p2; p1 in Hom(c,cod p1) & p2 in Hom(c,cod p2) by A1,A2,CAT_1:18; then consider i such that i in Hom(c,c) and A6: for k st k in Hom(c,c) holds p1*k = p1 & p2*k = p2 iff i = k by A3; id c in Hom(c,c) & p1*(id c) = p1 & p2*(id c) = p2 by A1,A2,CAT_1:47,55; hence id c = i by A6 .= h by A4,A5,A6; end; theorem c is_a_product_wrt p1,p2 & dom f = d & cod f = c & f is invertible implies d is_a_product_wrt p1*f,p2*f proof assume that A1: c is_a_product_wrt p1,p2 and A2: dom f = d & cod f = c & f is invertible; dom p1 = c & dom p2 = c by A1,Def16; then c is_a_product_wrt (0,1)-->(p1,p2) & (0,1)-->(p1,p2) is Projections_family of c,{0,1} by A1,Th48,Th59; then d is_a_product_wrt ((0,1)-->(p1,p2))*f by A2,Th56; then d is_a_product_wrt (0,1)-->(p1*f,p2*f) by Th18; hence d is_a_product_wrt p1*f,p2*f by Th59; end; theorem c is_a_product_wrt p1,p2 & cod p2 is terminal implies c,cod p1 are_isomorphic proof set a = cod p1, b = cod p2; assume that A1: c is_a_product_wrt p1,p2 and A2: b is terminal; A3: dom p1 = c & dom p2 = c by A1,Def16; hence Hom(c,a)<>{} by CAT_1:19; reconsider p = p1 as Morphism of c,a by A3,CAT_1:22; take p; set f = id(a),g = term(a,b); dom g = a & cod g = b by A2,Th39; then f in Hom(a,a) & g in Hom(a,b) by CAT_1:18,55; then consider h such that A4: h in Hom(a,c) and A5: for k st k in Hom(a,c) holds p1*k = f & p2*k = g iff h = k by A1,Def16; take h; thus dom h = cod p & cod h = dom p by A3,A4,CAT_1:18; thus A6: p*h = id cod p by A4,A5; now A7: dom h = a & cod h = c by A4,CAT_1:18; then A8: dom(h*p) = c & cod(h*p) = c by A3,CAT_1:42; hence h*p in Hom(c,c) by CAT_1:18; thus p1*(h*p) = (id cod p)*p by A3,A6,A7,CAT_1:43 .= p by CAT_1:46; dom(p2*(h*p)) = c & cod(p2*(h*p)) = b by A3,A8,CAT_1:42; hence p2*(h*p) = term(c,b) by A2,Th40 .= p2 by A2,A3,Th40; end; hence id dom p = h*p by A1,A3,Th63; end; theorem c is_a_product_wrt p1,p2 & cod p1 is terminal implies c,cod p2 are_isomorphic proof set a = cod p1, b = cod p2; assume that A1: c is_a_product_wrt p1,p2 and A2: a is terminal; A3: dom p1 = c & dom p2 = c by A1,Def16; hence Hom(c,b)<>{} by CAT_1:19; reconsider p = p2 as Morphism of c,b by A3,CAT_1:22; take p; set f = id(b),g = term(b,a); dom g = b & cod g = a by A2,Th39; then f in Hom(b,b) & g in Hom(b,a) by CAT_1:18,55; then consider h such that A4: h in Hom(b,c) and A5: for k st k in Hom(b,c) holds p1*k = g & p2*k = f iff h = k by A1,Def16; take h; thus dom h = cod p & cod h = dom p by A3,A4,CAT_1:18; thus A6: p*h = id cod p by A4,A5; now A7: dom h = b & cod h = c by A4,CAT_1:18; then A8: dom(h*p) = c & cod(h*p) = c by A3,CAT_1:42; hence h*p in Hom(c,c) by CAT_1:18; dom(p1*(h*p)) = c & cod(p1*(h*p)) = a by A3,A8,CAT_1:42; hence p1*(h*p) = term(c,a) by A2,Th40 .= p1 by A2,A3,Th40; thus p2*(h*p) = (id cod p)*p by A3,A6,A7,CAT_1:43 .= p by CAT_1:46; end; hence id dom p = h*p by A1,A3,Th63; end; begin :: Coproducts definition let C,c,I; mode Injections_family of c,I -> Function of I,the Morphisms of C means :Def17: cods it = I --> c; existence proof take F = I-->id c; cods F = I --> (cod id c) by Th9; hence thesis by CAT_1:44; end; end; theorem Th67: for F being Injections_family of c,I st x in I holds cod(F/.x) = c proof let F be Injections_family of c,I such that A1: x in I; (cods F)/.x = (I --> c)/.x by Def17; hence cod(F/.x) = (I --> c)/.x by A1,Def4 .= c by A1,Th2; end; theorem for F being Function of {},the Morphisms of C holds F is Injections_family of a,{} proof let F be Function of {},the Morphisms of C; thus {} --> a = {} by FUNCT_4:3 .= cods F by PARTFUN1:57; end; theorem Th69: cod f = a implies {y} --> f is Injections_family of a,{y} proof set F = {y} --> f; assume A1: cod f = a; now let x; assume A2: x in {y}; hence (cods F)/.x = cod(F/.x) by Def4 .= a by A1,A2,Th2 .= ({y} --> a)/.x by A2,Th2; end; hence cods F = {y} --> a by Th1; end; theorem Th70: cod p1 = c & cod p2 = c implies (x1,x2)-->(p1,p2) is Injections_family of c,{x1,x2} proof assume A1: cod p1 = c & cod p2 = c; cods ((x1,x2)-->(p1,p2)) = (x1,x2) --> (cod p1,cod p2) by Th11 .= {x1,x2} --> c by A1,FUNCT_4:68; hence thesis by Def17; end; canceled; theorem Th72: for F being Injections_family of b,I st dom f = b holds f*F is Injections_family of cod f,I proof let F be Injections_family of b,I; assume dom f = b; then cods F = I--> dom f by Def17; hence cods(f*F) = I --> cod f by Th21; end; theorem for F being Injections_family of b,I, G being Function of I,the Morphisms of C st doms F = cods G holds F"*"G is Injections_family of b,I proof let F be Injections_family of b,I; let G be Function of I,the Morphisms of C; assume doms F = cods G; then cods(F"*"G) = cods F by Th22; hence cods(F"*"G) = I --> b by Def17; end; theorem Th74: for F be Function of I,the Morphisms of C holds F is Projections_family of c,I iff F opp is Injections_family of c opp,I proof let F be Function of I,the Morphisms of C; thus F is Projections_family of c,I implies F opp is Injections_family of c opp,I proof assume A1: doms F = I --> c; now let x; assume A2: x in I; hence (cods(F opp))/.x = cod((F opp)/.x) by Def4 .= cod((F/.x) opp) by A2,Def5 .= dom(F/.x) opp by OPPCAT_1:11 .= ((I --> c)/.x) opp by A1,A2,Def3 .= c opp by A2,Th2 .= (I --> (c opp))/.x by A2,Th2; end; hence cods(F opp) = I --> (c opp) by Th1; end; assume A3: cods(F opp) = I --> (c opp); now let x; assume A4: x in I; hence (doms F)/.x = dom(F/.x) by Def3 .= cod((F/.x) opp) by OPPCAT_1:9 .= cod((F opp)/.x) by A4,Def5 .= (I --> (c opp))/.x by A3,A4,Def4 .= c opp by A4,Th2 .= c by OPPCAT_1:def 2 .= (I --> c)/.x by A4,Th2; end; hence doms F = I --> c by Th1; end; theorem Th75: for F be Function of I,the Morphisms of C opp, c being Object of C opp holds F is Injections_family of c,I iff opp F is Projections_family of opp c,I proof let F be Function of I,the Morphisms of C opp, c be Object of C opp; thus F is Injections_family of c,I implies opp F is Projections_family of opp c,I proof assume A1: cods F = I --> c; now let x; assume A2: x in I; hence (doms opp F)/.x = dom(opp F/.x) by Def3 .= dom(opp(F/.x)) by A2,Def6 .= opp(cod(F/.x)) by OPPCAT_1:12 .= opp((I-->c)/.x) by A1,A2,Def4 .= opp c by A2,Th2 .= (I --> (opp c))/.x by A2,Th2; end; hence doms opp F = I --> (opp c) by Th1; end; assume A3: doms opp F = I --> (opp c); now let x; assume A4: x in I; hence (cods F)/.x = cod(F/.x) by Def4 .= dom(opp(F/.x)) by OPPCAT_1:10 .= dom(opp F/.x) by A4,Def6 .= (I -->(opp c))/.x by A3,A4,Def3 .= opp c by A4,Th2 .= c opp by OPPCAT_1:def 3 .= c by OPPCAT_1:def 2 .= (I --> c)/.x by A4,Th2; end; hence cods F = I --> c by Th1; end; theorem for F being Injections_family of dom f,I holds (F opp)*(f opp) = (f*F) opp proof let F be Injections_family of dom f, I; now let x; assume A1: x in I; then A2: cod(F/.x) = (cods F)/.x by Def4 .= (I --> dom f)/.x by Def17 .= dom f by A1,Th2; thus ((F opp)*(f opp))/.x = ((F opp)/.x)*(f opp) by A1,Def7 .= ((F/.x)opp)*(f opp) by A1,Def5 .= (f*(F/.x))opp by A2,OPPCAT_1:17 .= ((f*F)/.x)opp by A1,Def8 .= ((f*F) opp)/.x by A1,Def5; end; hence thesis by Th1; end; definition let C,c,I; let F be Function of I,the Morphisms of C; pred c is_a_coproduct_wrt F means :Def18: F is Injections_family of c,I & for d for F' being Injections_family of d,I st doms F = doms F' ex h st h in Hom(c,d) & for k st k in Hom(c,d) holds (for x st x in I holds k*(F/.x) = F'/.x) iff h = k; end; theorem for F being Function of I,the Morphisms of C holds c is_a_product_wrt F iff c opp is_a_coproduct_wrt F opp proof let F be Function of I,the Morphisms of C; thus c is_a_product_wrt F implies c opp is_a_coproduct_wrt F opp proof assume A1: c is_a_product_wrt F; then F is Projections_family of c,I by Def15; hence F opp is Injections_family of c opp,I by Th74; let d be Object of C opp, F' be Injections_family of d,I such that A2: doms(F opp) = doms F'; reconsider oppF' = opp F' as Projections_family of opp d,I by Th75; now let x; assume A3: x in I; hence (cods F)/.x = cod(F/.x) by Def4 .= dom((F/.x)opp) by OPPCAT_1:9 .= dom(F opp/.x) by A3,Def5 .= (doms F')/.x by A2,A3,Def3 .= dom(F'/.x) by A3,Def3 .= cod(opp(F'/.x)) by OPPCAT_1:10 .= cod(oppF'/.x) by A3,Def6 .= (cods oppF')/.x by A3,Def4; end; then cods F = cods oppF' by Th1; then consider h such that A4: h in Hom(opp d,c) and A5: for k st k in Hom(opp d,c) holds (for x st x in I holds (F/.x)*k = oppF'/.x ) iff h = k by A1,Def15; take h opp; h opp in Hom(c opp,(opp d) opp) by A4,OPPCAT_1:13; hence h opp in Hom(c opp,d) by OPPCAT_1:5; let k be Morphism of C opp; assume A6: k in Hom(c opp,d); then opp k in Hom(opp d,opp(c opp)) by OPPCAT_1:14; then A7: opp k in Hom(opp d,c) by OPPCAT_1:4; thus (for x st x in I holds k*(F opp/.x) = F'/.x ) implies h opp = k proof assume A8: for x st x in I holds k*(F opp/.x) = F'/.x; now let x such that A9: x in I; F is Projections_family of c,I by A1,Def15; then dom(F/.x) = c by A9,Th45; then cod((F/.x)opp) = c opp by OPPCAT_1:11; then cod(F opp/.x) = c opp by A9,Def5; then A10: dom k = cod(F opp/.x) by A6,CAT_1:18; opp(k*(F opp/.x)) = opp(F'/.x) by A8,A9; hence oppF'/.x = opp(k*(F opp/.x)) by A9,Def6 .= (opp(F opp/.x))*(opp k) by A10,OPPCAT_1:19 .= (opp((F/.x)opp))*(opp k) by A9,Def5 .= (F/.x)*(opp k) by OPPCAT_1:7; end; then h = opp k by A5,A7; hence h opp = k by OPPCAT_1:8; end; assume A11: h opp = k; let x such that A12: x in I; F is Projections_family of c,I by A1,Def15; then dom(F/.x) = c by A12,Th45; then A13: cod opp k = dom(F/.x) by A7,CAT_1:18; h = opp k by A11,OPPCAT_1:7; then (F/.x)*(opp k) = oppF'/.x by A5,A7,A12; then ((opp k) opp)*((F/.x)opp) = (oppF'/.x)opp by A13,OPPCAT_1:17; then k*((F/.x)opp) = (oppF'/.x)opp by OPPCAT_1:8; hence k*(F opp/.x) = (oppF'/.x)opp by A12,Def5 .= (opp(F'/.x))opp by A12,Def6 .= F'/.x by OPPCAT_1:8; end; assume A14: c opp is_a_coproduct_wrt F opp; then F opp is Injections_family of c opp,I by Def18; hence F is Projections_family of c,I by Th74; let d; let F' be Projections_family of d,I such that A15: cods F = cods F'; reconsider F'opp = F' opp as Injections_family of d opp,I by Th74; now let x; assume A16: x in I; hence (doms(F opp))/.x = dom((F opp)/.x) by Def3 .= dom((F/.x) opp) by A16,Def5 .= cod(F/.x) by OPPCAT_1:9 .= (cods F')/.x by A15,A16,Def4 .= cod(F'/.x) by A16,Def4 .= dom((F'/.x)opp) by OPPCAT_1:9 .= dom((F' opp)/.x) by A16,Def5 .= (doms(F' opp))/.x by A16,Def3; end; then doms(F opp) = doms(F'opp) by Th1; then consider h being Morphism of C opp such that A17: h in Hom(c opp,d opp) and A18: for k being Morphism of C opp st k in Hom(c opp,d opp) holds (for x st x in I holds k*(F opp/.x) = F'opp/.x) iff h = k by A14,Def18; take opp h; opp h in Hom(opp(d opp),opp(c opp)) by A17,OPPCAT_1:14; then opp h in Hom(d,opp(c opp)) by OPPCAT_1:4; hence opp h in Hom(d,c) by OPPCAT_1:4; let k; assume A19: k in Hom(d,c); then A20: k opp in Hom(c opp,d opp) by OPPCAT_1:13; thus (for x st x in I holds (F/.x)*k = F'/.x) implies opp h = k proof assume A21: for x st x in I holds (F/.x)*k = F'/.x; now let x such that A22: x in I; F opp is Injections_family of c opp,I by A14,Def18; then cod(F opp/.x) = c opp by A22,Th67; then cod((F/.x)opp) = c opp by A22,Def5; then dom(F/.x) = c opp by OPPCAT_1:9; then dom(F/.x) = c by OPPCAT_1:def 2; then A23: cod k = dom(F/.x) by A19,CAT_1:18; (F/.x)*k = F'/.x by A21,A22; then (k opp)*((F/.x)opp) = (F'/.x) opp by A23,OPPCAT_1:17; hence F'opp/.x = (k opp)*((F/.x)opp) by A22,Def5 .= (k opp)*(F opp/.x) by A22,Def5; end; then h = k opp by A18,A20; hence opp h = k by OPPCAT_1:7; end; assume A24: opp h = k; let x such that A25: x in I; F opp is Injections_family of c opp,I by A14,Def18; then cod(F opp/.x) = c opp by A25,Th67; then cod((F/.x)opp) = c opp by A25,Def5; then dom(F/.x) = c opp by OPPCAT_1:9; then dom(F/.x) = c by OPPCAT_1:def 2; then A26: cod k = dom(F/.x) by A19,CAT_1:18; h = k opp by A24,OPPCAT_1:8; then (k opp)*(F opp/.x) = F'opp/.x by A18,A20,A25; then (k opp)*(F opp/.x) = (F'/.x)opp by A25,Def5; hence F'/.x = (k opp)*(F opp/.x) by OPPCAT_1:def 4 .= (k opp)*((F/.x)opp) by A25,Def5 .= ((F/.x)*k) opp by A26,OPPCAT_1:17 .= (F/.x)*k by OPPCAT_1:def 4; end; theorem Th78: for F being Injections_family of c,I, F' being Injections_family of d,I st c is_a_coproduct_wrt F & d is_a_coproduct_wrt F' & doms F = doms F' holds c,d are_isomorphic proof let F be Injections_family of c,I, F' be Injections_family of d,I such that A1: c is_a_coproduct_wrt F and A2: d is_a_coproduct_wrt F' and A3: doms F = doms F'; consider f such that A4: f in Hom(c,d) and A5: for k st k in Hom(c,d) holds (for x st x in I holds k*(F/.x) = F'/.x ) iff f = k by A1,A3,Def18; consider g such that A6: g in Hom(d,c) and A7: for k st k in Hom(d,c) holds (for x st x in I holds k*(F'/.x) = F/.x ) iff g = k by A2,A3,Def18; thus Hom(c,d) <> {} by A4; reconsider f as Morphism of c,d by A4,CAT_1:def 7; take f,g; A8: cod f = d & dom g = d & dom f = c & cod g = c by A4,A6,CAT_1:18; hence dom g = cod f & cod g = dom f; doms F' = doms F'; then consider fg being Morphism of C such that fg in Hom(d,d) and A9: for k st k in Hom(d,d) holds (for x st x in I holds k*(F'/.x) = F'/.x) iff fg = k by A2,Def18; A10: now set k = id d; thus k in Hom(d,d) by CAT_1:55; let x; assume x in I; then cod(F'/.x) = d by Th67; hence k*(F'/.x) = F'/.x by CAT_1:46; end; now cod(f*g) = d & dom(f*g) = d by A8,CAT_1:42; hence f*g in Hom(d,d) by CAT_1:18; let x; assume A11: x in I; then cod(F'/.x) = d by Th67; hence f*g*(F'/.x) = f*(g*(F'/.x)) by A8,CAT_1:43 .= f*(F/.x) by A6,A7,A11 .= F'/.x by A4,A5,A11; end; hence f*g = fg by A9 .= id cod f by A8,A9,A10; A12: now set k = id c; thus k in Hom(c,c) by CAT_1:55; let x; assume x in I; then cod(F/.x) = c by Th67; hence k*(F/.x) = F/.x by CAT_1:46; end; doms F = doms F; then consider gf being Morphism of C such that gf in Hom(c,c) and A13: for k st k in Hom(c,c) holds (for x st x in I holds k*(F/.x) = F/.x ) iff gf = k by A1,Def18; now cod(g*f) = c & dom(g*f) = c by A8,CAT_1:42; hence g*f in Hom(c,c) by CAT_1:18; let x; assume A14: x in I; then cod(F/.x) = c by Th67; hence g*f*(F/.x) = g*(f*(F/.x)) by A8,CAT_1:43 .= g*(F'/.x) by A4,A5,A14 .= F/.x by A6,A7,A14; end; hence g*f = gf by A13 .= id dom f by A8,A12,A13; end; theorem Th79: for F being Injections_family of c,I st c is_a_coproduct_wrt F & for x1,x2 st x1 in I & x2 in I holds Hom(dom(F/.x1),dom(F/.x2)) <> {} for x st x in I holds F/.x is coretraction proof let F be Injections_family of c,I such that A1: c is_a_coproduct_wrt F and A2: for x1,x2 st x1 in I & x2 in I holds Hom(dom(F/.x1),dom(F/.x2)) <> {}; let x such that A3: x in I; set d = dom(F/.x); defpred P[set,set] means ($1 = x implies $2 = id d) & ($1 <> x implies $2 in Hom(dom(F/.$1),d)); A4: for y st y in I holds ex z st z in the Morphisms of C & P[y,z] proof let y; assume y in I; then A5: Hom(dom(F/.y),d) <> {} by A2,A3; consider z being Element of Hom(dom(F/.y),d); A6: z is Morphism of dom(F/.y),d by A5,CAT_1:21; per cases; suppose A7: y = x; take z = id d; thus z in the Morphisms of C; thus (y = x implies z = id d) & (y <> x implies z in Hom(dom(F/.y),d)) by A7; suppose A8: y <> x; take z; thus z in the Morphisms of C by A6; thus (y = x implies z = id d) & (y <> x implies z in Hom(dom(F/.y),d)) by A5,A8; end; consider F' being Function such that A9: dom F' = I & rng F' c= the Morphisms of C and A10: for y st y in I holds P[y,F'.y] from NonUniqBoundFuncEx(A4); reconsider F' as Function of I,the Morphisms of C by A9,FUNCT_2:def 1, RELSET_1:11; now now let y; assume A11: y in I; then F'.y = F'/.y & F'.x = F'/.x by A3,Def1; then (y = x implies F'/.y = id d) & (y <> x implies F'/.y in Hom(dom(F/.y),d)) by A10,A11; then cod(F'/.y) = d by CAT_1:18,44; hence (cods F')/.y = d by A11,Def4 .= (I-->d)/.y by A11,Th2; end; hence cods F' = I --> d by Th1; hence F' is Injections_family of d,I by Def17; now let y; assume A12: y in I; then F'.y = F'/.y & F'.x = F'/.x by A3,Def1; then (y = x implies F'/.y = id d) & (y <> x implies F'/.y in Hom(dom(F/.y),d)) by A10,A12; then dom(F'/.y) = dom(F/.y) by CAT_1:18,44; hence (doms F')/.y = dom(F/.y) by A12,Def3 .= (doms F)/.y by A12,Def3; end; hence doms F = doms F' by Th1; end; then consider i such that A13: i in Hom(c,d) and A14: for k st k in Hom(c,d) holds (for y st y in I holds k*(F/.y) = F'/.y) iff i = k by A1,Def18; take i; thus dom i = c by A13,CAT_1:18 .= cod(F/.x) by A3,Th67; thus i*(F/.x) = F'/.x by A3,A13,A14 .= F'.x by A3,Def1 .= id d by A3,A10; end; theorem Th80: for F being Injections_family of a,I st a is_a_coproduct_wrt F & dom f = a & cod f = b & f is invertible holds b is_a_coproduct_wrt f*F proof let F be Injections_family of a,I such that A1: a is_a_coproduct_wrt F and A2: dom f = a & cod f = b and A3: f is invertible; thus f*F is Injections_family of b,I by A2,Th72; let c; let F' be Injections_family of c,I such that A4: doms(f*F) = doms F'; cods F = I-->dom f by A2,Def17; then doms F = doms F' by A4,Th21; then consider h such that A5: h in Hom(a,c) and A6: for k st k in Hom(a,c) holds (for x st x in I holds k*(F/.x) = F'/.x) iff h = k by A1,Def18; consider g such that A7: dom g = cod f & cod g = dom f and A8: f*g = id cod f and A9: g*f = id dom f by A3,CAT_1:def 12; take hg = h*g; A10: dom h = a & cod h = c by A5,CAT_1:18; then cod(h*g) = c & dom(h*g) = b by A2,A7,CAT_1:42; hence hg in Hom(b,c) by CAT_1:18; let k; assume k in Hom(b,c); then A11: cod k = c & dom k = b by CAT_1:18; thus (for x st x in I holds k*((f*F)/.x) = F'/.x) implies hg = k proof assume A12: for x st x in I holds k*((f*F)/.x) = F'/.x; now cod(k*f) = c & dom(k*f) = a by A2,A11,CAT_1:42; hence k*f in Hom(a,c) by CAT_1:18; let x; assume A13: x in I; then cod(F/.x) = a by Th67; hence k*f*(F/.x) = k*(f*(F/.x)) by A2,A11,CAT_1:43 .= k*((f*F)/.x) by A13,Def8 .= F'/.x by A12,A13; end; then k*f*g = h*g by A6; hence hg = k*(id b) by A2,A7,A8,A11,CAT_1:43 .= k by A11,CAT_1:47; end; assume A14: hg = k; let x; assume A15: x in I; then A16: cod(F/.x) = a by Th67; then A17: cod(f*(F/.x)) = b by A2,CAT_1:42; thus k*((f*F)/.x) = k*(f*(F/.x)) by A15,Def8 .= h*(g*(f*(F/.x))) by A2,A7,A10,A14,A17,CAT_1:43 .= h*((id dom f)*(F/.x)) by A2,A7,A9,A16,CAT_1:43 .= h*(F/.x) by A2,A16,CAT_1:46 .= F'/.x by A5,A6,A15; end; theorem Th81: for F being Injections_family of a,{} holds a is_a_coproduct_wrt F iff a is initial proof let F be Injections_family of a,{}; thus a is_a_coproduct_wrt F implies a is initial proof assume A1: a is_a_coproduct_wrt F; let b; consider F' being Injections_family of b,{}; doms F = {} & doms F' = {} by PARTFUN1:57; then consider h such that A2: h in Hom(a,b) and A3: for k st k in Hom(a,b) holds (for x st x in {} holds k*(F/.x) = F'/.x ) iff h = k by A1,Def18; thus Hom(a,b)<>{} by A2; reconsider f = h as Morphism of a,b by A2,CAT_1:def 7; take f; let g be Morphism of a,b; g in Hom(a,b) & for x st x in {} holds g*(F/.x) = F'/.x by A2,CAT_1:def 7; hence f=g by A3; end; assume A4: a is initial; thus F is Injections_family of a,{}; let b; let F' be Injections_family of b,{} such that doms F = doms F'; consider h being Morphism of a,b such that A5: for g being Morphism of a,b holds h = g by A4,CAT_1:def 16; take h; Hom(a,b)<>{} by A4,CAT_1:def 16; hence h in Hom(a,b) by CAT_1:def 7; let k; assume k in Hom(a,b); then k is Morphism of a,b by CAT_1:def 7; hence (for x st x in {} holds k*(F/.x) = F'/.x ) iff h = k by A5; end; theorem a is_a_coproduct_wrt {y} --> id a proof set F = {y} --> id a; cod(id a) = a by CAT_1:44; hence F is Injections_family of a,{y} by Th69; let b; let F' be Injections_family of b,{y} such that A1: doms F = doms F'; take h = F'/.y; A2: y in {y} by TARSKI:def 1; now thus cod h = b by A2,Th67; thus dom h = (doms F)/.y by A1,A2,Def3 .= dom(F/.y) by A2,Def3 .= dom(id a) by A2,Th2 .= a by CAT_1:44; end; hence h in Hom(a,b) by CAT_1:18; let k; assume k in Hom(a,b); then A3: dom k = a by CAT_1:18; thus (for x st x in {y} holds k*(F/.x) = F'/.x) implies h = k proof assume A4: for x st x in {y} holds k*(F/.x) = F'/.x; thus k = k*(id a) by A3,CAT_1:47 .= k*(F/.y) by A2,Th2 .= h by A2,A4; end; assume A5: h = k; let x; assume A6: x in {y}; hence F'/.x = k by A5,TARSKI:def 1 .= k*(id a) by A3,CAT_1:47 .= k*(F/.x) by A6,Th2; end; theorem for F being Injections_family of a,I st a is_a_coproduct_wrt F & for x st x in I holds dom(F/.x) is initial holds a is initial proof let F be Injections_family of a,I such that A1: a is_a_coproduct_wrt F and A2: for x st x in I holds dom(F/.x) is initial; now thus I = {} implies a is initial by A1,Th81; let b; deffunc f(set) = init(dom(F/.$1),b); consider F' being Function of I, the Morphisms of C such that A3: for x st x in I holds F'/.x = f(x) from LambdaIdx; now let x; assume A4: x in I; then A5: dom(F/.x) is initial by A2; thus (cods F')/.x = cod(F'/.x) by A4,Def4 .= cod(init(dom(F/.x),b)) by A3,A4 .= b by A5,Th42 .= (I-->b)/.x by A4,Th2; end; then cods F' = I --> b by Th1; then reconsider F' as Injections_family of b,I by Def17; now let x; assume A6: x in I; then A7: dom(F/.x) is initial by A2; thus (doms F)/.x = dom(F/.x) by A6,Def3 .= dom(init(dom(F/.x),b)) by A7,Th42 .= dom(F'/.x) by A3,A6 .= (doms F')/.x by A6,Def3; end; then doms F = doms F' by Th1; then consider h such that A8: h in Hom(a,b) and A9: for k st k in Hom(a,b) holds (for x st x in I holds k*(F/.x) = F'/.x) iff h = k by A1,Def18; thus Hom(a,b)<>{} by A8; reconsider h as Morphism of a,b by A8,CAT_1:def 7; take h; let g be Morphism of a,b; now thus g in Hom(a,b) by A8,CAT_1:def 7; let x; set c = dom(F/.x); assume A10: x in I; then c is initial by A2; then consider f being Morphism of c,b such that A11: for f' being Morphism of c,b holds f = f' by CAT_1:def 16; cod g = b & dom g = a & cod(F/.x) = a by A8,A10,Th67,CAT_1:23; then cod(g*(F/.x)) = b & dom(g*(F/.x)) = c by CAT_1:42; then g*(F/.x) in Hom(c,b) by CAT_1:18; then A12: g*(F/.x) is Morphism of c,b by CAT_1:def 7; F'/.x = init(c,b) by A3,A10; hence F'/.x = f by A11 .= g*(F/.x) by A11,A12; end; hence h = g by A9; end; hence thesis by CAT_1:def 16; end; definition let C,c,i1,i2; pred c is_a_coproduct_wrt i1,i2 means :Def19: cod i1 = c & cod i2 = c & for d,f,g st f in Hom(dom i1,d) & g in Hom(dom i2,d) ex h st h in Hom(c,d) & for k st k in Hom(c,d) holds k*i1 = f & k*i2 = g iff h = k; end; theorem c is_a_product_wrt p1,p2 iff c opp is_a_coproduct_wrt p1 opp, p2 opp proof set i1 = p1 opp,i2 = p2 opp; thus c is_a_product_wrt p1,p2 implies c opp is_a_coproduct_wrt p1 opp, p2 opp proof assume that A1: dom p1 = c & dom p2 = c and A2: for d,f,g st f in Hom(d,cod p1) & g in Hom(d,cod p2) ex h st h in Hom(d,c) & for k st k in Hom(d,c) holds p1*k = f & p2*k = g iff h = k; dom p1 = c opp & dom p2 = c opp by A1,OPPCAT_1:def 2; hence A3: cod i1 = c opp & cod i2 = c opp by OPPCAT_1:9; let d be Object of C opp, f,g be Morphism of C opp; assume f in Hom(dom i1,d) & g in Hom(dom i2,d); then opp f in Hom(opp d, opp(dom i1)) & opp g in Hom(opp d,opp(dom i2)) by OPPCAT_1:14; then opp f in Hom(opp d, cod opp i1) & opp g in Hom(opp d,cod opp i2) by OPPCAT_1:12; then opp f in Hom(opp d, cod p1) & opp g in Hom(opp d,cod p2) by OPPCAT_1:7; then consider h such that A4: h in Hom(opp d,c) and A5: for k st k in Hom(opp d,c) holds p1*k = opp f & p2*k = opp g iff h = k by A2; take h opp; h opp in Hom(c opp,(opp d) opp) by A4,OPPCAT_1:13; hence h opp in Hom(c opp,d) by OPPCAT_1:5; let k be Morphism of C opp; assume A6: k in Hom(c opp,d); then opp k in Hom(opp d,opp(c opp)) by OPPCAT_1:14; then opp k in Hom(opp d,c) by OPPCAT_1:4; then p1*(opp k) = opp f & p2*(opp k) = opp g iff h = opp k by A5; then p1*(opp k) = f opp & p2*(opp k) = g opp iff h = k opp by OPPCAT_1:def 5; then p1*(opp k) = f & p2*(opp k) = g iff h = k by OPPCAT_1:def 4; then ((opp i1)*(opp k) = f & (opp i2)*(opp k) = g iff h opp = k) & dom k = c opp by A6,CAT_1:18,OPPCAT_1:7,def 4; then opp(k*i1) = f & opp(k*i2) = g iff h opp = k by A3,OPPCAT_1:19; then (k*i1)opp = f & (k*i2)opp = g iff h opp = k by OPPCAT_1:def 5; hence k*i1 = f & k*i2 = g iff h opp = k by OPPCAT_1:def 4; end; assume that A7: cod i1 = c opp & cod i2 = c opp and A8: for d being Object of C opp, f,g being Morphism of C opp st f in Hom(dom i1,d) & g in Hom(dom i2,d) ex h being Morphism of C opp st h in Hom(c opp,d) & for k being Morphism of C opp st k in Hom(c opp,d) holds k*i1 = f & k*i2 = g iff h = k; dom p1 = c opp & dom p2 = c opp by A7,OPPCAT_1:9; hence A9: dom p1 = c & dom p2 = c by OPPCAT_1:def 2; let d,f,g; assume f in Hom(d,cod p1) & g in Hom(d,cod p2); then f opp in Hom((cod p1) opp,d opp) & g opp in Hom((cod p2)opp,d opp) by OPPCAT_1:13; then f opp in Hom(dom (p1 opp),d opp) & g opp in Hom(dom (p2 opp),d opp) by OPPCAT_1:11; then consider h being Morphism of C opp such that A10: h in Hom(c opp,d opp) and A11: for k being Morphism of C opp st k in Hom(c opp,d opp) holds k*i1 = f opp & k*i2 = g opp iff h = k by A8; take opp h; h = (opp h) opp by OPPCAT_1:8; hence opp h in Hom(d,c) by A10,OPPCAT_1:13; let k; assume A12: k in Hom(d,c); then k opp in Hom(c opp,d opp) by OPPCAT_1:13; then (k opp)*i1 = f opp & (k opp)*i2 = g opp iff h = k opp by A11; then ((k opp)*i1 = f & (k opp)*i2 = g iff h opp = k opp) & cod k = c by A12,CAT_1:18,OPPCAT_1:def 4; then (p1*k) opp = f & (p2*k) opp = g iff h opp = k by A9,OPPCAT_1:17,def 4; hence thesis by OPPCAT_1:def 4,def 5; end; theorem Th85: x1 <> x2 implies (c is_a_coproduct_wrt i1,i2 iff c is_a_coproduct_wrt (x1,x2)-->(i1,i2)) proof set F = (x1,x2)-->(i1,i2), I = {x1,x2}; assume A1: x1 <> x2; thus c is_a_coproduct_wrt i1,i2 implies c is_a_coproduct_wrt F proof assume A2: c is_a_coproduct_wrt i1,i2; then cod i1 = c & cod i2 = c by Def19; hence F is Injections_family of c,I by Th70; let b; let F' be Injections_family of b,I such that A3: doms F = doms F'; set f = F'/.x1, g = F'/.x2; A4: x1 in I & x2 in I by TARSKI:def 2; then (doms F)/.x1 = dom(F/.x1) & (doms F)/.x2 = dom(F/.x2) by Def3; then dom f = dom(F/.x1) & dom g = dom(F/.x2) by A3,A4,Def3; then dom f = dom i1 & dom g = dom i2 & cod f = b & cod g = b by A1,A4,Th7,Th67; then f in Hom(dom i1,b) & g in Hom(dom i2,b) by CAT_1:18; then consider h such that A5: h in Hom(c,b) and A6: for k st k in Hom(c,b) holds k*i1 = f & k*i2 = g iff h = k by A2,Def19; take h; thus h in Hom(c,b) by A5; let k such that A7: k in Hom(c,b); thus (for x st x in I holds k*(F/.x) = F'/.x ) implies h = k proof assume for x st x in I holds k*(F/.x) = F'/.x; then k*(F/.x1) = f & k*(F/.x2) = g by A4; then k*i1 = f & k*i2 = g by A1,Th7; hence h = k by A6,A7; end; assume A8: h = k; let x such that A9: x in I; k*i1 = f & k*i2 = g by A6,A7,A8; then k*(F/.x1) = f & k*(F/.x2) = g & (x = x1 or x = x2) by A1,A9,Th7,TARSKI:def 2; hence thesis; end; assume A10: c is_a_coproduct_wrt F; then x1 in I & x2 in I & F is Injections_family of c,I by Def18,TARSKI:def 2 ; then cod(F/.x1) = c & cod(F/.x2) = c by Th67; hence cod i1 = c & cod i2 = c by A1,Th7; let d,f,g such that A11: f in Hom(dom i1,d) and A12: g in Hom(dom i2,d); cod f = d & cod g = d by A11,A12,CAT_1:18; then reconsider F' = (x1,x2) --> (f,g) as Injections_family of d,I by Th70; doms F = (x1,x2)-->(dom i1,dom i2) by Th10 .= (x1,x2)-->(dom f,dom i2) by A11,CAT_1:18 .= (x1,x2)-->(dom f,dom g) by A12,CAT_1:18 .= doms F' by Th10; then consider h such that A13: h in Hom(c,d) and A14: for k st k in Hom(c,d) holds (for x st x in I holds k*(F/.x) = F'/.x ) iff h = k by A10,Def18; take h; thus h in Hom(c,d) by A13; let k such that A15: k in Hom(c,d); thus k*i1 = f & k*i2 = g implies h = k proof assume A16: k*i1 = f & k*i2 = g; now let x; assume x in I; then x = x1 or x = x2 by TARSKI:def 2; then F/.x = i1 & F'/.x = f or F/.x = i2 & F'/.x = g by A1,Th7; hence k*(F/.x) = F'/.x by A16; end; hence h = k by A14,A15; end; assume h = k; then x1 in I & x2 in I & for x st x in I holds k*(F/.x) = F'/.x by A14,A15,TARSKI:def 2; then k*(F/.x1) = F'/.x1 & k*(F/.x2) = F'/.x2; then k*(F/.x1) = f & k*(F/.x2) = g by A1,Th7; hence k*i1 = f & k*i2 = g by A1,Th7; end; theorem c is_a_coproduct_wrt i1,i2 & d is_a_coproduct_wrt j1,j2 & dom i1 = dom j1 & dom i2 = dom j2 implies c,d are_isomorphic proof assume that A1: c is_a_coproduct_wrt i1,i2 and A2: d is_a_coproduct_wrt j1,j2 and A3: dom i1 = dom j1 & dom i2 = dom j2; set I = {0,1}, F = (0,1)-->(i1,i2), F' = (0,1)-->(j1,j2); now cod i1 = c & cod i2 = c & cod j1 = d & cod j2 = d by A1,A2,Def19; hence F is Injections_family of c,I & F' is Injections_family of d,I by Th70; thus c is_a_coproduct_wrt F & d is_a_coproduct_wrt F' by A1,A2,Th85; thus doms F = (0,1)-->(dom j1,dom j2) by A3,Th10 .= doms F' by Th10; end; hence thesis by Th78; end; theorem Hom(a,c) <> {} & Hom(b,c) <> {} implies for i1 being Morphism of a,c, i2 being Morphism of b,c holds c is_a_coproduct_wrt i1,i2 iff for d st Hom(a,d)<>{} & Hom(b,d)<>{} holds Hom(c,d) <> {} & for f being Morphism of a,d, g being Morphism of b,d ex h being Morphism of c,d st for k being Morphism of c,d holds k*i1 = f & k*i2 = g iff h = k proof assume A1: Hom(a,c) <> {} & Hom(b,c) <> {}; let i1 be Morphism of a,c, i2 be Morphism of b,c; thus c is_a_coproduct_wrt i1,i2 implies for d st Hom(a,d)<>{} & Hom(b,d)<>{} holds Hom(c,d) <> {} & for f being Morphism of a,d, g being Morphism of b,d ex h being Morphism of c,d st for k being Morphism of c,d holds k*i1 = f & k*i2 = g iff h = k proof assume that cod i1 = c & cod i2 = c and A2: for d,f,g st f in Hom(dom i1,d) & g in Hom(dom i2,d) ex h st h in Hom(c,d) & for k st k in Hom(c,d) holds k*i1 = f & k*i2 = g iff h = k; let d such that A3: Hom(a,d)<>{} & Hom(b,d)<>{}; consider f being Morphism of a,d, g being Morphism of b,d; A4: dom i1 = a & dom i2 = b by A1,CAT_1:23; then f in Hom(dom i1,d) & g in Hom(dom i2,d) by A3,CAT_1:def 7; then A5: ex h st h in Hom(c,d) & for k st k in Hom(c,d) holds k*i1 = f & k*i2 = g iff h = k by A2; hence Hom(c,d) <> {}; let f be Morphism of a,d, g be Morphism of b,d; f in Hom(dom i1,d) & g in Hom(dom i2,d) by A3,A4,CAT_1:def 7; then consider h such that A6: h in Hom(c,d) and A7: for k st k in Hom(c,d) holds k*i1 = f & k*i2 = g iff h = k by A2; reconsider h as Morphism of c,d by A6,CAT_1:def 7; take h; let k be Morphism of c,d; k*i1 = k*(i1 qua Morphism of C) & k*i2 = k*(i2 qua Morphism of C) & k in Hom(c,d) by A1,A5,CAT_1:def 7,def 13; hence thesis by A7; end; assume A8: for d st Hom(a,d)<>{} & Hom(b,d)<>{} holds Hom(c,d) <> {} & for f being Morphism of a,d, g being Morphism of b,d ex h being Morphism of c,d st for k being Morphism of c,d holds k*i1 = f & k*i2 = g iff h = k; thus cod i1 = c & cod i2 = c by A1,CAT_1:23; let d,f,g such that A9: f in Hom(dom i1,d) and A10: g in Hom(dom i2,d); A11: dom i1 = a & dom i2 = b by A1,CAT_1:23; then f is Morphism of a,d & g is Morphism of b,d & Hom(a,d) <> {} & Hom(b,d) <> {} by A9,A10,CAT_1:def 7; then consider h being Morphism of c,d such that A12: for k being Morphism of c,d holds k*i1 = f & k*i2 = g iff h = k by A8; reconsider h' = h as Morphism of C; take h'; A13: Hom(c,d) <> {} by A8,A9,A10,A11; hence h' in Hom(c,d) by CAT_1:def 7; let k; assume k in Hom(c,d); then reconsider k' = k as Morphism of c,d by CAT_1:def 7; k*i1 = k'*i1 & k*i2 = k'*i2 by A1,A13,CAT_1:def 13; hence thesis by A12; end; theorem c is_a_coproduct_wrt i1,i2 & Hom(dom i1,dom i2) <> {} & Hom(dom i2,dom i1) <> {} implies i1 is coretraction & i2 is coretraction proof assume that A1: c is_a_coproduct_wrt i1,i2 and A2: Hom(dom i1,dom i2) <> {} & Hom(dom i2,dom i1) <> {}; set I = {0,1}, F = (0,1)-->(i1,i2); A3: F/.0 = i1 & F/.1 = i2 by Th7; now cod i1 = c & cod i2 = c by A1,Def19; hence F is Injections_family of c,I by Th70; thus c is_a_coproduct_wrt F by A1,Th85; let x1,x2; assume x1 in I & x2 in I; then (x1 = 0 or x1 = 1) & (x2 = 0 or x2 = 1) by TARSKI:def 2; hence Hom(dom(F/.x1),dom(F/.x2)) <> {} by A2,A3,CAT_1:56; end; then 0 in I & 1 in I & for x st x in I holds F/.x is coretraction by Th79,TARSKI:def 2; hence thesis by A3; end; theorem Th89: c is_a_coproduct_wrt i1,i2 & h in Hom(c,c) & h*i1 = i1 & h*i2 = i2 implies h = id c proof assume that A1: cod i1 = c and A2: cod i2 = c and A3: for d,f,g st f in Hom(dom i1,d) & g in Hom(dom i2,d) ex h st h in Hom(c,d) & for k st k in Hom(c,d) holds k*i1 = f & k*i2 = g iff h = k and A4: h in Hom(c,c) and A5: h*i1 = i1 & h*i2 = i2; i1 in Hom(dom i1,c) & i2 in Hom(dom i2,c) by A1,A2,CAT_1:18; then consider i such that i in Hom(c,c) and A6: for k st k in Hom(c,c) holds k*i1 = i1 & k*i2 = i2 iff i = k by A3; id c in Hom(c,c) & (id c)*i1 = i1 & (id c)*i2 = i2 by A1,A2,CAT_1:46,55; hence id c = i by A6 .= h by A4,A5,A6; end; theorem c is_a_coproduct_wrt i1,i2 & dom f = c & cod f = d & f is invertible implies d is_a_coproduct_wrt f*i1,f*i2 proof assume that A1: c is_a_coproduct_wrt i1,i2 and A2: dom f = c & cod f = d & f is invertible; cod i1 = c & cod i2 = c by A1,Def19; then c is_a_coproduct_wrt (0,1)-->(i1,i2) & (0,1)-->(i1,i2) is Injections_family of c,{0,1} by A1,Th70,Th85; then d is_a_coproduct_wrt f*((0,1)-->(i1,i2)) by A2,Th80; then d is_a_coproduct_wrt (0,1)-->(f*i1,f*i2) by Th19; hence d is_a_coproduct_wrt f*i1,f*i2 by Th85; end; theorem c is_a_coproduct_wrt i1,i2 & dom i2 is initial implies dom i1,c are_isomorphic proof set a = dom i1, b = dom i2; assume that A1: c is_a_coproduct_wrt i1,i2 and A2: b is initial; A3: cod i1 = c & cod i2 = c by A1,Def19; hence Hom(a,c)<>{} by CAT_1:19; reconsider i = i1 as Morphism of a,c by A3,CAT_1:22; take i; set f = id(a),g = init(b,a); cod g = a & dom g = b by A2,Th42; then f in Hom(a,a) & g in Hom(b,a) by CAT_1:18,55; then consider h such that A4: h in Hom(c,a) and A5: for k st k in Hom(c,a) holds k*i1 = f & k*i2 = g iff h = k by A1,Def19; take h; thus dom h = cod i & cod h = dom i by A3,A4,CAT_1:18; now A6: cod h = a & dom h = c by A4,CAT_1:18; then A7: cod(i*h) = c & dom(i*h) = c by A3,CAT_1:42; hence i*h in Hom(c,c) by CAT_1:18; thus i*h*i1 = i*(h*i1) by A3,A6,CAT_1:43 .= i*(id dom i) by A4,A5 .= i by CAT_1:47; cod(i*h*i2) = c & dom(i*h*i2) = b by A3,A7,CAT_1:42; hence i*h*i2 = init(b,c) by A2,Th43 .= i2 by A2,A3,Th43; end; hence i*h = id cod i by A1,A3,Th89; thus id dom i = h*i by A4,A5; end; theorem c is_a_coproduct_wrt i1,i2 & dom i1 is initial implies dom i2,c are_isomorphic proof set a = dom i1, b = dom i2; assume that A1: c is_a_coproduct_wrt i1,i2 and A2: a is initial; A3: cod i1 = c & cod i2 = c by A1,Def19; hence Hom(b,c)<>{} by CAT_1:19; reconsider i = i2 as Morphism of b,c by A3,CAT_1:22; take i; set f = id(b),g = init(a,b); cod g = b & dom g = a by A2,Th42; then f in Hom(b,b) & g in Hom(a,b) by CAT_1:18,55; then consider h such that A4: h in Hom(c,b) and A5: for k st k in Hom(c,b) holds k*i1 = g & k*i2 = f iff h = k by A1,Def19; take h; thus dom h = cod i & cod h = dom i by A3,A4,CAT_1:18; now A6: cod h = b & dom h = c by A4,CAT_1:18; then A7: cod(i*h) = c & dom(i*h) = c by A3,CAT_1:42; hence i*h in Hom(c,c) by CAT_1:18; cod(i*h*i1) = c & dom(i*h*i1) = a by A3,A7,CAT_1:42; hence i*h*i1 = init(a,c) by A2,Th43 .= i1 by A2,A3,Th43; thus i*h*i2 = i*(h*i2) by A3,A6,CAT_1:43 .= i*(id dom i) by A4,A5 .= i by CAT_1:47; end; hence i*h = id cod i by A1,A3,Th89; thus id dom i = h*i by A4,A5; end;

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