let I be set ; :: thesis: for C being Category
for c being Object of C
for F being Injections_family of c,I st c is_a_coproduct_wrt F & ( for x1, x2 being set st x1 in I & x2 in I holds
Hom ((dom (F /. x1)),(dom (F /. x2))) <> {} ) holds
for x being set st x in I holds
for d being Object of C st d = dom (F /. x) & Hom (d,c) <> {} holds
for f being Morphism of d,c st f = F /. x holds
f is coretraction
let C be Category; :: thesis: for c being Object of C
for F being Injections_family of c,I st c is_a_coproduct_wrt F & ( for x1, x2 being set st x1 in I & x2 in I holds
Hom ((dom (F /. x1)),(dom (F /. x2))) <> {} ) holds
for x being set st x in I holds
for d being Object of C st d = dom (F /. x) & Hom (d,c) <> {} holds
for f being Morphism of d,c st f = F /. x holds
f is coretraction
let c be Object of C; :: thesis: for F being Injections_family of c,I st c is_a_coproduct_wrt F & ( for x1, x2 being set st x1 in I & x2 in I holds
Hom ((dom (F /. x1)),(dom (F /. x2))) <> {} ) holds
for x being set st x in I holds
for d being Object of C st d = dom (F /. x) & Hom (d,c) <> {} holds
for f being Morphism of d,c st f = F /. x holds
f is coretraction
let F be Injections_family of c,I; :: thesis: ( c is_a_coproduct_wrt F & ( for x1, x2 being set st x1 in I & x2 in I holds
Hom ((dom (F /. x1)),(dom (F /. x2))) <> {} ) implies for x being set st x in I holds
for d being Object of C st d = dom (F /. x) & Hom (d,c) <> {} holds
for f being Morphism of d,c st f = F /. x holds
f is coretraction )
assume that
A1: c is_a_coproduct_wrt F and
A2: for x1, x2 being set st x1 in I & x2 in I holds
Hom ((dom (F /. x1)),(dom (F /. x2))) <> {} ; :: thesis: for x being set st x in I holds
for d being Object of C st d = dom (F /. x) & Hom (d,c) <> {} holds
for f being Morphism of d,c st f = F /. x holds
f is coretraction
let x be set ; :: thesis: ( x in I implies for d being Object of C st d = dom (F /. x) & Hom (d,c) <> {} holds
for f being Morphism of d,c st f = F /. x holds
f is coretraction )
assume A3: x in I ; :: thesis: for d being Object of C st d = dom (F /. x) & Hom (d,c) <> {} holds
for f being Morphism of d,c st f = F /. x holds
f is coretraction
let d be Object of C; :: thesis: ( d = dom (F /. x) & Hom (d,c) <> {} implies for f being Morphism of d,c st f = F /. x holds
f is coretraction )
assume that
A4: d = dom (F /. x) and
A5: Hom (d,c) <> {} ; :: thesis: for f being Morphism of d,c st f = F /. x holds
f is coretraction
let f be Morphism of d,c; :: thesis: ( f = F /. x implies f is coretraction )
assume A6: f = F /. x ; :: thesis: f is coretraction
defpred S₁[ set , set ] means ( ( $1 = x implies $2 = id d ) & ( $1 <> x implies $2 in Hom ((dom (F /. $1)),d) ) );
A7: for y being set st y in I holds
ex z being set st
( z in the carrier' of C & S₁[y,z] )

proof

let y be set ; :: thesis: ( y in I implies ex z being set st
( z in the carrier' of C & S₁[y,z] ) )
set z = the Element of Hom ((dom (F /. y)),d);
assume y in I ; :: thesis: ex z being set st
( z in the carrier' of C & S₁[y,z] )
then A8: Hom ((dom (F /. y)),d) <> {} by A2, A3, A4;
then A9: the Element of Hom ((dom (F /. y)),d) in Hom ((dom (F /. y)),d) ;

per cases ( y = x or y <> x ) ;

suppose A10: y = x ; :: thesis: ex z being set st
( z in the carrier' of C & S₁[y,z] )

take z = id d; :: thesis: ( z in the carrier' of C & S₁[y,z] )
thus z in the carrier' of C ; :: thesis: S₁[y,z]
thus S₁[y,z] by A10; :: thesis: verum

end;

suppose A11: y <> x ; :: thesis: ex z being set st
( z in the carrier' of C & S₁[y,z] )

take the Element of Hom ((dom (F /. y)),d) ; :: thesis: ( the Element of Hom ((dom (F /. y)),d) in the carrier' of C & S₁[y, the Element of Hom ((dom (F /. y)),d)] )
thus the Element of Hom ((dom (F /. y)),d) in the carrier' of C by A9; :: thesis: S₁[y, the Element of Hom ((dom (F /. y)),d)]
thus S₁[y, the Element of Hom ((dom (F /. y)),d)] by A8, A11; :: thesis: verum

end;

end;

end;

consider F9 being Function such that
A12: ( dom F9 = I & rng F9 c= the carrier' of C ) and
A13: for y being set st y in I holds
S₁[y,F9 . y] from FUNCT_1:sch 5(A7);
reconsider F9 = F9 as Function of I, the carrier' of C by A12, FUNCT_2:def 1, RELSET_1:4;

now :: thesis: for y being set st y in I holds
(cods F9) /. y = (I --> d) /. y

let y be set ; :: thesis: ( y in I implies (cods F9) /. y = (I --> d) /. y )
assume A14: y in I ; :: thesis: (cods F9) /. y = (I --> d) /. y
then A15: F9 . y = F9 /. y by FUNCT_2:def 13;
then A16: ( y <> x implies F9 /. y in Hom ((dom (F /. y)),d) ) by A13, A14;
( y = x implies F9 /. y = id d ) by A13, A14, A15;
then cod (F9 /. y) = d by A16, CAT_1:1, CAT_1:58;
hence (cods F9) /. y = d by A14, Def2
.= (I --> d) /. y by A14, Th2 ;
:: thesis: verum

end;

then A17: F9 is Injections_family of d,I by Def16, Th1;

now :: thesis: for y being set st y in I holds
(doms F9) /. y = (doms F) /. y

let y be set ; :: thesis: ( y in I implies (doms F9) /. y = (doms F) /. y )
assume A18: y in I ; :: thesis: (doms F9) /. y = (doms F) /. y
then A19: F9 . y = F9 /. y by FUNCT_2:def 13;
then A20: ( y <> x implies F9 /. y in Hom ((dom (F /. y)),d) ) by A13, A18;
( y = x implies F9 /. y = id d ) by A13, A18, A19;
then dom (F9 /. y) = dom (F /. y) by A20, A4, CAT_1:1, CAT_1:58;
hence (doms F9) /. y = dom (F /. y) by A18, Def1
.= (doms F) /. y by A18, Def1 ;
:: thesis: verum

end;

then consider i being Morphism of C such that
A21: i in Hom (c,d) and
A22: for k being Morphism of C st k in Hom (c,d) holds
( ( for y being set st y in I holds
k (*) (F /. y) = F9 /. y ) iff i = k ) by A1, A17, Def17, Th1;
thus ( Hom (d,c) <> {} & Hom (c,d) <> {} ) by A21, A5; :: according to CAT_3:def 9 :: thesis: ex g being Morphism of c,d st g * f = id d
reconsider i = i as Morphism of c,d by A21, CAT_1:def 5;
take i ; :: thesis: i * f = id d
thus i * f = i (*) (F /. x) by A6, A21, A5, CAT_1:def 13
.= F9 /. x by A3, A21, A22
.= F9 . x by A3, FUNCT_2:def 13
.= id d by A3, A13 ; :: thesis: verum