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 Th32:
  G1 is_naturally_transformable_to G2 implies t*id B = t
proof
  assume
A1: G1 is_naturally_transformable_to G2;
A2: G1*id B = G1 by FUNCT_2:17;
  then reconsider s = t*id B as natural_transformation of G1,G2 by FUNCT_2:17;
A3: G2*id B = G2 by FUNCT_2:17;
  now
    let b be Object of B;
    thus s.b = t.((id B).b) by A1,A2,A3,Th22
      .= t.b by CAT_1:79;
  end;
  hence thesis by A1,Th24;
end;
