reserve B,C,D for Category;

theorem
  for S being Contravariant_Functor of C opp,B holds /*S is Functor of C , B
proof
  let S be Contravariant_Functor of C opp,B;
  now
    thus for c being Object of C ex d being Object of B st /*S.(id c) = id d
    proof
      let c be Object of C;
      (/*S).(id c) = id ((Obj /*S).c) by Lm11;
      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 Lm11
        .= id dom (/*S.f) by Lm12;
      thus (/*S).(id cod f) = id((Obj /*S).(cod f)) by Lm11
        .= id cod (/*S.f) by Lm12;
    end;
    let f,g be Morphism of C such that
A1: dom g = cod f;
A2: dom(f opp) = cod f & cod (g opp) = dom g;

     reconsider ff=f as Morphism of dom f,cod f by CAT_1:4;
     reconsider gg=g as Morphism of cod f,cod g by A1,CAT_1:4;
A3:   Hom(dom f,cod f)<>{} & Hom(dom g,cod g)<>{} by CAT_1:2;
     then
A4:   ff opp = f opp by Def6;
A5:   gg opp = g opp by Def6,A3,A1;
    thus /*S.(g(*)f) = S.((g(*)f) opp) by Def8
      .= S.((f opp)(*)(g opp)) by A4,A5,A3,A1,Th14
      .= (S.(g opp))(*)(S.(f opp)) by A1,A2,Def9
      .= (/*S.g)(*)(S.(f opp)) by Def8
      .= (/*S.g)(*)(/*S.f) by Def8;
  end;
  hence thesis by CAT_1:61;
end;
