
theorem Th62:
  for C,D being category, F,F1,F2 being Functor of C,D
  st F1 is_naturally_transformable_to F & F is_naturally_transformable_to F2 &
  F is covariant & F1 is covariant & F2 is covariant
  holds F1 is_naturally_transformable_to F2
  proof
    let C,D be category;
    let F,F1,F2 be Functor of C,D;
    set T1 = the natural_transformation of F1,F;
    set T2 = the natural_transformation of F,F2;
    assume F1 is_naturally_transformable_to F &
    F is_naturally_transformable_to F2 & F is covariant & F1 is covariant
    & F2 is covariant;
    then ex T being natural_transformation of F1,F2 st
    T is_natural_transformation_of F1,F2 &
    for f,f1,f2 being morphism of C st f1 is identity & f2 is identity &
    f |> f1 & f2 |> f holds T.f = (T2.f2)(*)(F.f)(*)(T1.f1) by Lm3;
    hence F1 is_naturally_transformable_to F2;
  end;
