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 Th29:
  F1 is_naturally_transformable_to F2 implies H*G*s = H*(G*s)
proof
  assume
A1: F1 is_naturally_transformable_to F2;
A2: H*(G*F1) = H*G*F1 by RELAT_1:36;
  then reconsider v = H*(G*s) as natural_transformation of H*G*F1, H*G*F2 by
RELAT_1:36;
A3: H*(G*F2) = H*G*F2 by RELAT_1:36;
  now
    let a be Object of A;
A4: G.(F1.a) = (G*F1).a & G.(F2.a) = (G*F2).a by CAT_1:76;
A5: Hom(F1.a,F2.a) <> {} by A1,Th23;
    thus (H*G*s).a = (H*G)/.(s.a) by A1,Th21
      .= H/.(G/.(s.a)) by A5,NATTRA_1:11
      .= H/.((G*s).a) by A1,A4,Th21
      .= v.a by A1,A2,A3,Th20,Th21;
  end;
  hence thesis by A1,Th20,Th24;
end;
