let C be Category; :: thesis: for a being Object of C
for F being Injections_family of a, {} holds
( a is_a_coproduct_wrt F iff a is initial )
let a be Object of C; :: thesis: for F being Injections_family of a, {} holds
( a is_a_coproduct_wrt F iff a is initial )
let F be Injections_family of a, {} ; :: thesis: ( a is_a_coproduct_wrt F iff a is initial )
thus ( a is_a_coproduct_wrt F implies a is initial ) :: thesis: ( a is initial implies a is_a_coproduct_wrt F )

proof

assume A1: a is_a_coproduct_wrt F ; :: thesis: a is initial
let b be Object of C; :: according to CAT_1:def 13 :: thesis: ( not Hom (a,b) = {} & ex b₁ being Morphism of a,b st
for b₂ being Morphism of a,b holds b₁ = b₂ )
set F9 = the Injections_family of b, {} ;
doms F = {} ;
then consider h being Morphism of C such that
A2: h in Hom (a,b) and
A3: for k being Morphism of C st k in Hom (a,b) holds
( ( for x being set st x in {} holds
k * (F /. x) = the Injections_family of b, {} /. x ) iff h = k ) by A1, Def18;
thus Hom (a,b) <> {} by A2; :: thesis: ex b₁ being Morphism of a,b st
for b₂ being Morphism of a,b holds b₁ = b₂
reconsider f = h as Morphism of a,b by A2, CAT_1:def 4;
take f ; :: thesis: for b₁ being Morphism of a,b holds f = b₁
let g be Morphism of a,b; :: thesis: f = g
A4: for x being set st x in {} holds
g * (F /. x) = the Injections_family of b, {} /. x ;
g in Hom (a,b) by A2, CAT_1:def 4;
hence f = g by A3, A4; :: thesis: verum

end;

assume A5: a is initial ; :: thesis: a is_a_coproduct_wrt F
thus F is Injections_family of a, {} ; :: according to CAT_3:def 16 :: thesis: for d being Object of C
for F9 being Injections_family of d, {} st doms F = doms F9 holds
ex h being Morphism of C st
( h in Hom (a,d) & ( for k being Morphism of C st k in Hom (a,d) holds
( ( for x being set st x in {} holds
k * (F /. x) = F9 /. x ) iff h = k ) ) )
let b be Object of C; :: thesis: for F9 being Injections_family of b, {} st doms F = doms F9 holds
ex h being Morphism of C st
( h in Hom (a,b) & ( for k being Morphism of C st k in Hom (a,b) holds
( ( for x being set st x in {} holds
k * (F /. x) = F9 /. x ) iff h = k ) ) )
consider h being Morphism of a,b such that
A6: for g being Morphism of a,b holds h = g by A5, CAT_1:def 13;
let F9 be Injections_family of b, {} ; :: thesis: ( doms F = doms F9 implies ex h being Morphism of C st
( h in Hom (a,b) & ( for k being Morphism of C st k in Hom (a,b) holds
( ( for x being set st x in {} holds
k * (F /. x) = F9 /. x ) iff h = k ) ) ) )
assume doms F = doms F9 ; :: thesis: ex h being Morphism of C st
( h in Hom (a,b) & ( for k being Morphism of C st k in Hom (a,b) holds
( ( for x being set st x in {} holds
k * (F /. x) = F9 /. x ) iff h = k ) ) )
take h ; :: thesis: ( h in Hom (a,b) & ( for k being Morphism of C st k in Hom (a,b) holds
( ( for x being set st x in {} holds
k * (F /. x) = F9 /. x ) iff h = k ) ) )
Hom (a,b) <> {} by A5, CAT_1:def 13;
hence h in Hom (a,b) by CAT_1:def 4; :: thesis: for k being Morphism of C st k in Hom (a,b) holds
( ( for x being set st x in {} holds
k * (F /. x) = F9 /. x ) iff h = k )
let k be Morphism of C; :: thesis: ( k in Hom (a,b) implies ( ( for x being set st x in {} holds
k * (F /. x) = F9 /. x ) iff h = k ) )
assume k in Hom (a,b) ; :: thesis: ( ( for x being set st x in {} holds
k * (F /. x) = F9 /. x ) iff h = k )
then k is Morphism of a,b by CAT_1:def 4;
hence ( ( for x being set st x in {} holds
k * (F /. x) = F9 /. x ) iff h = k ) by A6; :: thesis: verum