reserve A,B,C for Category,
  F,F1,F2,F3 for Functor of A,B,
  G for Functor of B, C;
reserve m,o for set;

theorem Th17:
  F1 is_transformable_to F2 implies for t being transformation of
  F1,F2 holds (id F2)`*`t = t & t`*`(id F1) = t
proof
  assume
A1: F1 is_transformable_to F2;
  let t be transformation of F1,F2;
  now
    let a be Object of A;
A2: Hom(F1.a,F2.a) <> {} by A1;
    thus ((id F2)`*`t).a = ((id F2).a)*(t.a) by A1,Def5
      .= (id(F2.a))*(t.a) by Th16
      .= t.a by A2,CAT_1:28;
  end;
  hence (id F2)`*`t = t by A1,Th15;
  now
    let a be Object of A;
A3: Hom(F1.a,F2.a) <> {} by A1;
    thus (t`*`(id F1)).a = (t.a)*((id F1).a) by A1,Def5
      .= (t.a)*(id(F1.a)) by Th16
      .= t.a by A3,CAT_1:29;
  end;
  hence thesis by A1,Th15;
end;
