let V be non empty set ; :: thesis: for C being Category
for a, c being Object of C st Hom C c= V holds
for d being Object of (Ens V) st d = Hom a,c holds
(hom?- a) . (id c) = id d
let C be Category; :: thesis: for a, c being Object of C st Hom C c= V holds
for d being Object of (Ens V) st d = Hom a,c holds
(hom?- a) . (id c) = id d
let a, c be Object of C; :: thesis: ( Hom C c= V implies for d being Object of (Ens V) st d = Hom a,c holds
(hom?- a) . (id c) = id d )
A1: Hom a,c in Hom C ;
assume Hom C c= V ; :: thesis: for d being Object of (Ens V) st d = Hom a,c holds
(hom?- a) . (id c) = id d
then reconsider A = Hom a,c as Element of V by A1;
A2: ( dom (id c) = c & cod (id c) = c ) by CAT_1:44;
A3: hom a,(id c) = id A by Th43;
let d be Object of (Ens V); :: thesis: ( d = Hom a,c implies (hom?- a) . (id c) = id d )
assume d = Hom a,c ; :: thesis: (hom?- a) . (id c) = id d
hence (hom?- a) . (id c) = id$ (@ d) by A2, A3, Def22
.= id d by Def13 ;
:: thesis: verum