let B, C be Category; :: thesis: for S being Contravariant_Functor of C,B
for c being Object of C holds (S *') . (id c) = id ((Obj (S *')) . c)
let S be Contravariant_Functor of C,B; :: thesis: for c being Object of C holds (S *') . (id c) = id ((Obj (S *')) . c)
let c be Object of C; :: thesis: (S *') . (id c) = id ((Obj (S *')) . c)
A1: Hom (((Obj S) . c),((Obj S) . c)) <> {} ;
thus (S *') . (id c) = (S . (id c)) opp by Def11
.= (id ((Obj S) . c)) opp by Th29
.= (id ((Obj S) . c)) opp by Def6, A1
.= id (((Obj S) . c) opp) by Th18
.= id ((Obj (S *')) . c) by Th43 ; :: thesis: verum