let V be non empty set ; :: thesis: for C being Category
for a, b being Object of C st Hom C c= V holds
(Obj (hom-? (V,a))) . b = Hom (b,a)
let C be Category; :: thesis: for a, b being Object of C st Hom C c= V holds
(Obj (hom-? (V,a))) . b = Hom (b,a)
let a, b be Object of C; :: thesis: ( Hom C c= V implies (Obj (hom-? (V,a))) . b = Hom (b,a) )
assume A1: Hom C c= V ; :: thesis: (Obj (hom-? (V,a))) . b = Hom (b,a)
Hom (b,a) in Hom C ;
then reconsider A = Hom (b,a) as Element of V by A1;
set d = @ A;
(hom-? (V,a)) . (id b) = (hom-? a) . (id b) by A1, Def25
.= id (@ A) by A1, Lm10 ;
hence (Obj (hom-? (V,a))) . b = Hom (b,a) by OPPCAT_1:30; :: thesis: verum