let C, B be Category; :: thesis: for S being Functor of C,B
for c being Object of C holds (Obj (S *' )) . c = ((Obj S) . c) opp
let S be Functor of C,B; :: thesis: for c being Object of C holds (Obj (S *' )) . c = ((Obj S) . c) opp
let c be Object of C; :: thesis: (Obj (S *' )) . c = ((Obj S) . c) opp

now

thus (S *' ) . (id c) = id (((Obj S) . c) opp ) by Lm11; :: thesis: for c being Object of C ex d being Object of (B opp ) st (S *' ) . (id c) = id d
let c be Object of C; :: thesis: ex d being Object of (B opp ) st (S *' ) . (id c) = id d
(S *' ) . (id c) = id (((Obj S) . c) opp ) by Lm11;
hence ex d being Object of (B opp ) st (S *' ) . (id c) = id d ; :: thesis: verum

end;

hence (Obj (S *' )) . c = ((Obj S) . c) opp by CAT_1:102; :: thesis: verum