reserve B,C,D for Category;

theorem Th56:
  for S being Functor of C,D holds S*' is Contravariant_Functor of C, D opp
proof
  let S be Functor of C,D;
  thus for c being Object of C ex d being Object of D opp st S*'.(id c) = id d
  proof
    let c be Object of C;
    (S*').(id c) = id ((Obj S*').c) by Lm25;
    hence thesis;
  end;
  thus for f being Morphism of C holds S*'.(id dom f) = id cod (S*'.f) & S*'.(
  id cod f) = id dom (S*'.f)
  proof
    let f be Morphism of C;
    thus (S*').(id dom f) = id((Obj S*').(dom f)) by Lm25
      .= id cod(S*'.f) by Lm26;
    thus (S*').(id cod f) = id((Obj S*').(cod f)) by Lm25
      .= id dom(S*'.f) by Lm26;
  end;
  let f,g be Morphism of C;
  assume
A1: dom g = cod f;
  then
A2: dom(S.g) = cod (S.f) by CAT_1:64;
     reconsider Sff=S.f as Morphism of dom(S.f),cod(S.f) by CAT_1:4;
     reconsider Sgg=S.g as Morphism of dom(S.g),cod(S.g) by CAT_1:4;
A3:   Hom(dom(S.f),cod(S.f))<>{} & Hom(dom(S.g),cod(S.g))<>{} by CAT_1:2;
     then
A4:   Sff opp = (S.f)opp by Def6;
A5:   Sgg opp = (S.g)opp by Def6,A3;
  thus S*'.(g(*)f) = (S.(g(*)f)) opp by Def11
    .= ((Sgg)(*)(Sff)) opp by A1,CAT_1:64
    .= ((Sff) opp)(*)((Sgg) opp) by A2,Th14,A3
    .= (S*'.f)(*)((S.g) opp) by Def11,A4,A5
    .= (S*'.f)(*)(S*'.g) by Def11;
end;
