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 Th34:
  F1 is_naturally_transformable_to F2 & G1
  is_naturally_transformable_to G2 implies t(#)s = (G2*s)`*`(t*F1)
proof
  assume that
A1: F1 is_naturally_transformable_to F2 and
A2: G1 is_naturally_transformable_to G2;
A3: G1*F1 is_naturally_transformable_to G1*F2 & G1*F2
  is_naturally_transformable_to G2*F2 by A1,A2,Th20;
A4: G2*F1 is_naturally_transformable_to G2*F2 & G1*F1
  is_naturally_transformable_to G2*F1 by A1,A2,Th20;
  now
    let a be Object of A;
A5: (G1*F1).a = G1.(F1.a) & (G1*F2).a = G1.(F2.a) by CAT_1:76;
A6: (G2*F2).a = G2.(F2.a) by CAT_1:76;
A7: (G2*s).a = G2/.(s.a) & (G1*F1).a = G1.(F1.a) by A1,Th21,CAT_1:76;
A8: Hom(F1.a,F2.a) <> {} & (t*F1).a = t.(F1.a) by A1,A2,Th22,Th23;
A9: (G2*F1).a = G2.(F1.a) & (G2*F2).a = G2.(F2.a) by CAT_1:76;
    (t*F2).a = t.(F2.a) & (G1*s).a = G1/.(s.a) by A1,A2,Th21,Th22;
    hence ((t*F2)`*`(G1*s)).a = t.(F2.a)*G1/.(s.a) by A3,A5,A6,NATTRA_1:25
      .= (G2*s).a*(t*F1).a by A2,A8,A7,A9,NATTRA_1:def 8
      .= ((G2*s)`*`(t*F1)).a by A4,NATTRA_1:25;
  end;
  hence thesis by A3,Th24,NATTRA_1:23;
end;
