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 Th42:
  for F being Functor of A,B, G being Functor of B,A for I being
  Functor of A,A st I ~= id A holds F*I ~= F & I*G ~= G
proof
  let F be Functor of A,B, G be Functor of B,A;
  let I be Functor of A,A;
  F*id A = F & id A*G = G by FUNCT_2:17;
  hence thesis by Th41;
end;
