reserve B,C,D for Category;

theorem Th32:
  for S being Functor of C opp,B holds /*S is Contravariant_Functor of C,B
proof
  let S be Functor of C opp,B;
  thus for c being Object of C ex d being Object of B st /*S.(id c) = id d
     by Lm6;
  thus for f being Morphism of C holds /*S.(id dom f) = id cod (/*S.f) &
         /*S.(id cod f) = id dom (/*S.f) by Lm8;
    let f,g be Morphism of C such that
A1:  dom g = cod f;
     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;
     Hom(dom f,cod f)<>{} & Hom(dom g,cod g)<>{} by CAT_1:2;
     then /*S.(gg(*)ff) = (/*S.ff)(*)(/*S.gg) by A1,Lm9;
    hence thesis;
end;
