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 Th27:
  H1 is_naturally_transformable_to H2 implies u*G*F = u*(G*F)
proof
  assume
A1: H1 is_naturally_transformable_to H2;
A2: H1*(G*F) = H1*G*F by RELAT_1:36;
  then reconsider v = u*(G*F) as natural_transformation of H1*G*F, H2*G*F by
RELAT_1:36;
A3: H2*(G*F) = H2*G*F by RELAT_1:36;
A4: now
    let a be Object of A;
    thus (u*G*F).a = (u*G).(F.a) by A1,Th20,Th22
      .= u.(G.(F.a)) by A1,Th22
      .= u.((G*F).a) by CAT_1:76
      .= v.a by A1,A2,A3,Th22;
  end;
  H1*G is_naturally_transformable_to H2*G by A1,Th20;
  hence thesis by A4,Th20,Th24;
end;
