reserve B,C,D for Category;

theorem Th54:
  for S being Contravariant_Functor of C,D holds S*' is Functor of C, D opp
proof
  let S be Contravariant_Functor of C,D;
  now
    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) opp) by Lm16;
      hence thesis;
    end;
    thus for f being Morphism of C holds S*'.(id dom f) = id dom (S*'.f) & S*'
    .(id cod f) = id cod (S*'.f)
    proof
      let f be Morphism of C;
      thus (S*').(id dom f) = id((Obj S*').(dom f)) by Lm21
        .= id dom (S*'.f) by Lm22;
      thus (S*').(id cod f) = id((Obj S*').(cod f)) by Lm21
        .= id cod (S*'.f) by Lm22;
    end;
    let f,g be Morphism of C;
    assume
A1: dom g = cod f;
    then
A2: dom(S.f) = cod (S.g) by Th31;
     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 A3,Def6;
    thus S*'.(g(*)f) = (S.(g(*)f)) opp by Def11
      .= ((Sff)(*)(Sgg)) opp by A1,Def9
      .= ((Sgg) opp)(*)((Sff) opp) by A3,A2,Th14
      .= (S*'.g)(*)((S.f) opp) by Def11,A4,A5
      .= (S*'.g)(*)(S*'.f) by Def11;
  end;
  hence thesis by CAT_1:61;
end;
