let C be Category; :: thesis:
let a, c, b be Object of C; :: thesis:
assume that
A1: Hom a,c <> {} and
A2: Hom b,c <> {} ; :: thesis:
let i1 be Morphism of a,c; :: thesis:
let i2 be Morphism of b,c; :: thesis:
thus ( c is_a_coproduct_wrt i1,i2 implies for d being Object of C st Hom a,d <> {} & Hom b,d <> {} holds
( Hom c,d <> {} & ( for f being Morphism of a,d
for 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 ) ) ) ) :: thesis:

proof

assume that
cod i1 = c and
cod i2 = c and
A3: for d being Object of C
for f, g being Morphism of C st f in Hom (dom i1),d & g in Hom (dom i2),d holds
ex h being Morphism of C st
( h in Hom c,d & ( for k being Morphism of C st k in Hom c,d holds
( ( k * i1 = f & k * i2 = g ) iff h = k ) ) ) ; :: according to CAT_3:def 19 :: thesis:
let d be Object of C; :: thesis:
assume that
A4: Hom a,d <> {} and
A5: Hom b,d <> {} ; :: thesis:
consider f being Morphism of a,d, g being Morphism of b,d;
A6: dom i2 = b by A2, CAT_1:23;
then A7: g in Hom (dom i2),d by A5, CAT_1:def 7;
A8: dom i1 = a by A1, CAT_1:23;
then f in Hom (dom i1),d by A4, CAT_1:def 7;
then A9: ex h being Morphism of C st
( h in Hom c,d & ( for k being Morphism of C st k in Hom c,d holds
( ( k * i1 = f & k * i2 = g ) iff h = k ) ) ) by A3, A7;
hence Hom c,d <> {} ; :: thesis:
let f be Morphism of a,d; :: thesis:
let g be Morphism of b,d; :: thesis:
A10: g in Hom (dom i2),d by A5, A6, CAT_1:def 7;
f in Hom (dom i1),d by A4, A8, CAT_1:def 7;
then consider h being Morphism of C such that
A11: h in Hom c,d and
A12: for k being Morphism of C st k in Hom c,d holds
( ( k * i1 = f & k * i2 = g ) iff h = k ) by A3, A10;
reconsider h = h as Morphism of c,d by A11, CAT_1:def 7;
take h ; :: thesis:
let k be Morphism of c,d; :: thesis:
A13: k in Hom c,d by A9, CAT_1:def 7;
( k * i1 = k * i1 & k * i2 = k * i2 ) by A1, A2, A9, CAT_1:def 13;
hence ( ( k * i1 = f & k * i2 = g ) iff h = k ) by A12, A13; :: thesis:

end;

assume A14: for d being Object of C st Hom a,d <> {} & Hom b,d <> {} holds
( Hom c,d <> {} & ( for f being Morphism of a,d
for 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 ) ) ) ; :: thesis:
thus ( cod i1 = c & cod i2 = c ) by A1, A2, CAT_1:23; :: according to CAT_3:def 19 :: thesis:
let d be Object of C; :: thesis:
let f, g be Morphism of C; :: thesis:
assume that
A15: f in Hom (dom i1),d and
A16: g in Hom (dom i2),d ; :: thesis:
A17: Hom a,d <> {} by A1, A15, CAT_1:23;
A18: dom i1 = a by A1, CAT_1:23;
then A19: f is Morphism of a,d by A15, CAT_1:def 7;
A20: dom i2 = b by A2, CAT_1:23;
then g is Morphism of b,d by A16, CAT_1:def 7;
then consider h being Morphism of c,d such that
A21: for k being Morphism of c,d holds
( ( k * i1 = f & k * i2 = g ) iff h = k ) by A14, A16, A20, A19, A17;
reconsider h' = h as Morphism of C ;
take h' ; :: thesis:
A22: Hom c,d <> {} by A14, A15, A16, A18, A20;
hence h' in Hom c,d by CAT_1:def 7; :: thesis:
let k be Morphism of C; :: thesis:
assume k in Hom c,d ; :: thesis:
then reconsider k' = k as Morphism of c,d by CAT_1:def 7;
( k * i1 = k' * i1 & k * i2 = k' * i2 ) by A1, A2, A22, CAT_1:def 13;
hence ( ( k * i1 = f & k * i2 = g ) iff h' = k ) by A21; :: thesis: