reserve B,C,D for Category;

theorem Th31:
  for S being Contravariant_Functor of C,D, f,g being Morphism of
  C st dom g = cod f holds dom (S.f) = cod (S.g)
proof
  let S be Contravariant_Functor of C,D, f,g be Morphism of C;
  assume dom g = cod f;
  hence dom (S.f) = (Obj S).(dom g) by Th30
    .= cod (S.g) by Th30;
end;
