reserve A,B,C,D for Category,
  F for Functor of A,B,
  G for Functor of B,C;
reserve o,m for set;

theorem Th22:
  for G1,G2 be Functor of B,C st G1 is_naturally_transformable_to
  G2 for F be Functor of A,B, t be natural_transformation of G1,G2, a be Object
  of A holds (t*F).a = t.(F.a)
proof
  let G1,G2 be Functor of B,C;
  assume
A1: G1 is_naturally_transformable_to G2;
  then
A2: G1 is_transformable_to G2;
  let F be Functor of A,B, t be natural_transformation of G1,G2, a be Object
  of A;
  thus (t*F).a = ((t qua transformation of G1,G2)*F).a by A1,Def8
    .= t.(F.a) by A2,Th18;
end;
