reserve B,C,D for Category;

theorem Th55:
  for S being Functor of C,D holds *'S is Contravariant_Functor of C opp,D
proof
  let S be Functor of C,D;
  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 Lm23;
    hence thesis;
  end;
  thus for f being Morphism of C opp holds *'S.(id dom f) = id cod (*'S.f) &
  *'S.(id cod f) = id dom (*'S.f)
  proof
    let f be Morphism of C opp;
    thus (*'S).(id dom f) = id((Obj *'S).(dom f)) by Lm23
      .= id cod(*'S.f) by Lm24;
    thus (*'S).(id cod f) = id((Obj *'S).(cod f)) by Lm23
      .= id dom(*'S.f) by Lm24;
  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 f))(*)(S.(opp g)) by A1,A2,CAT_1:64
    .= (*'S.f)(*)(S.(opp g)) by Def10
    .= (*'S.f)(*)(*'S.g) by Def10;
end;
