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 Th31:
  G*id F = id(G*F)
proof
  now
    let a be Object of A;
    thus (G*id F).a = G/.(id F.a) by Th21
      .= G/.id(F.a) by NATTRA_1:20
      .= id(G.(F.a)) by NATTRA_1:15
      .= id((G*F).a) by CAT_1:76
      .= (id(G*F)).a by NATTRA_1:20;
  end;
  hence thesis by Th24;
end;
