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 Th25:
  F1 is_naturally_transformable_to F2 & F2
  is_naturally_transformable_to F3 implies G*(s9`*`s) = (G*s9)`*`(G*s)
proof
  assume
A1: F1 is_naturally_transformable_to F2 & F2 is_naturally_transformable_to F3;
  then
A2: G*F1 is_naturally_transformable_to G*F2 & G*F2
  is_naturally_transformable_to G*F3 by Th20;
  now
    let a be Object of A;
A3: G.(F1.a) = (G*F1).a & G.(F2.a) = (G*F2).a by CAT_1:76;
A4: G.(F3.a) = (G*F3).a by CAT_1:76;
A5: Hom(F1.a,F2.a) <> {} & Hom(F2.a,F3.a) <> {} by A1,Th23;
A6: G/.(s9.a) = (G*s9).a & G/.(s.a) = (G*s).a by A1,Th21;
    thus (G*(s9`*`s)).a = G/.((s9`*`s).a) by A1,Th21,NATTRA_1:23
      .= G/.((s9.a)*(s.a)) by A1,NATTRA_1:25
      .= G/.(s9.a)*G/.(s.a) by A5,NATTRA_1:13
      .= ((G*s9)`*`(G*s)).a by A2,A6,A3,A4,NATTRA_1:25;
  end;
  hence thesis by A2,Th24,NATTRA_1:23;
end;
