reserve B,C,D for Category;

theorem Th41:
  for S being Contravariant_Functor of C,B, c being Object of C
  opp holds (Obj *'S).c = (Obj S).(opp c)
proof
  let S be Contravariant_Functor of C,B, c be Object of C opp;
  now
    thus (*'S).(id c) = id((Obj S).(opp c)) by Lm15;
    let c be Object of C opp;
    (*'S).(id c) = id((Obj S).(opp c)) by Lm15;
    hence ex d being Object of B st (*'S).(id c) = id d;
  end;
  hence thesis by CAT_1:66;
end;
