reserve B,C,D for Category;

theorem Th53:
  for S being Contravariant_Functor of C,D holds *'S is Functor of C opp,D
proof
  let S be Contravariant_Functor of C,D;
  now
    thus for c being Object of C opp ex d being Object of D st *'S.(id c) = id
    d
    proof
      let c be Object of C opp;
      (*'S).(id c) = id ((Obj *'S).c) by Lm19;
      hence thesis;
    end;
    thus for f being Morphism of C opp holds *'S.(id dom f) = id dom (*'S.f) &
    *'S.(id cod f) = id cod (*'S.f)
    proof
      let f be Morphism of C opp;
      thus (*'S).(id dom f) = id((Obj *'S).(dom f)) by Lm19
        .= id dom (*'S.f) by Lm20;
      thus (*'S).(id cod f) = id((Obj *'S).(cod f)) by Lm19
        .= id cod (*'S.f) by Lm20;
    end;
    let f,g be Morphism of C opp such that
A1: dom g = cod f;
A2: dom(opp f) = cod f & cod (opp g) = dom g;
    thus *'S.(g(*)f) = S.(opp (g(*)f)) by Def10
      .= S.((opp f)(*)(opp g)) by A1,Th16
      .= (S.(opp g))(*)(S.(opp f)) by A1,A2,Def9
      .= (*'S.g)(*)(S.(opp f)) by Def10
      .= (*'S.g)(*)(*'S.f) by Def10;
  end;
  hence thesis by CAT_1:61;
end;
