
theorem Th61:
  for C,D being category, F being Functor of C,D
  st F is covariant holds F is_natural_transformation_of F,F
  proof
    let C,D be category;
    let F be Functor of C,D;
    assume
A1: F is covariant;
    then
A2: F is multiplicative by CAT_6:def 25;
    for f,f1,f2 being morphism of C st f1 is identity & f2 is identity &
    f1 |> f & f |> f2 holds F.f1 |> F.f & F.f |> F.f2 &
    F.f = (F.f1)(*)(F.f) & F.f = (F.f)(*)(F.f2)
    proof
      let f,f1,f2 be morphism of C;
      assume
A3:  f1 is identity;
      assume
A4:  f2 is identity;
      assume
A5:   f1 |> f;
      assume
A6:   f |> f2;
      thus F.f1 |> F.f by A2,A5,CAT_6:def 23;
      thus F.f |> F.f2 by A2,A6,CAT_6:def 23;
      thus F.f = F.(f1(*)f) by A3,A5,CAT_6:def 4,def 14
      .= (F.f1)(*)(F.f) by A2,A5,CAT_6:def 23;
      thus F.f = F.(f(*)f2) by A4,A6,CAT_6:def 5,def 14
      .= (F.f)(*)(F.f2) by A2,A6,CAT_6:def 23;
    end;
    hence F is_natural_transformation_of F,F by A1,Th58;
  end;
