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 Th28:
  G1 is_naturally_transformable_to G2 implies H*t*F = H*(t*F)
proof
  assume
A1: G1 is_naturally_transformable_to G2;
A2: H*(G1*F) = H*G1*F by RELAT_1:36;
  then reconsider v = H*(t*F) as natural_transformation of H*G1*F, H*G2*F by
RELAT_1:36;
A3: H*(G2*F) = H*G2*F by RELAT_1:36;
A4: now
    let a be Object of A;
A5: G1.(F.a) = (G1*F).a & G2.(F.a) = (G2*F).a by CAT_1:76;
    thus (H*t*F).a = (H*t).(F.a) by A1,Th20,Th22
      .= H/.(t.(F.a)) by A1,Th21
      .= H/.((t*F).a) by A1,A5,Th22
      .= v.a by A1,A2,A3,Th20,Th21;
  end;
  H*G1 is_naturally_transformable_to H*G2 by A1,Th20;
  hence thesis by A4,Th20,Th24;
end;
