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,C))) . [(a opp),b] = Hom (a,b)
let C be Category; :: thesis: for a, b being Object of C st Hom C c= V holds
(Obj (hom?? (V,C))) . [(a opp),b] = Hom (a,b)
let a, b be Object of C; :: thesis: ( Hom C c= V implies (Obj (hom?? (V,C))) . [(a opp),b] = Hom (a,b) )
assume A1: Hom C c= V ; :: thesis: (Obj (hom?? (V,C))) . [(a opp),b] = Hom (a,b)
Hom (a,b) in Hom C ;
then reconsider A = Hom (a,b) as Element of V by A1;
A2: id (a opp) = id a by OPPCAT_1:71;
set d = @ A;
(hom?? (V,C)) . (id [(a opp),b]) = (hom?? (V,C)) . [(id (a opp)),(id b)] by CAT_2:31
.= (hom?? C) . [(id a),(id b)] by A1, Def26, A2
.= id (@ A) by A1, Lm11 ;
hence (Obj (hom?? (V,C))) . [(a opp),b] = Hom (a,b) by CAT_1:67; :: thesis: verum