reserve B,C,D for Category;

theorem
  for S being Function of the carrier' of C,the carrier' of B for T
  being Function of the carrier' of B,the carrier' of D holds (T*S)*' = T*'*S
proof
  let S be Function of the carrier' of C,the carrier' of B;
  let T be Function of the carrier' of B,the carrier' of D;
  now
    let f be Morphism of C;
    thus (T*S)*'.f = ((T*S).f) opp by Def11
      .= (T.(S.f)) opp by FUNCT_2:15
      .= T*'.(S.f) by Def11
      .= (T*'*S).f by FUNCT_2:15;
  end;
  hence thesis by FUNCT_2:63;
end;
