reserve A,B,C,D for Category,
  F for Functor of A,B,
  G for Functor of B,C;
reserve o,m for set;
reserve F,F1,F2,F3 for Functor of A,B,
  G,G1,G2,G3 for Functor of B,C,
  H,H1,H2 for Functor of C,D,
  s for natural_transformation of F1,F2,
  s9 for natural_transformation of F2,F3,
  t for natural_transformation of G1,G2,
  t9 for natural_transformation of G2,G3,
  u for natural_transformation of H1,H2;

theorem Th33:
  F1 is_naturally_transformable_to F2 implies (id B)*s = s
proof
  assume
A1: F1 is_naturally_transformable_to F2;
A2: (id B)*F1 = F1 by FUNCT_2:17;
  then reconsider t = (id B)*s as natural_transformation of F1,F2 by FUNCT_2:17
;
A3: (id B)*F2 = F2 by FUNCT_2:17;
  now
    let a be Object of A;
A4: Hom(F1.a,F2.a) <> {} by A1,Th23;
    thus t.a = (id B)/.(s.a) by A1,A2,A3,Th21
      .= (id B).(s.a qua Morphism of B) by A4,CAT_3:def 10
      .= s.a by FUNCT_1:18;
  end;
  hence thesis by A1,Th24;
end;
