reserve B,C,D for Category;

theorem
  for S being Function of the carrier' of C,the carrier' of D holds *'S
  = S*(*id C) & S*' = (id* D)*S
proof
  let S be Function of the carrier' of C,the carrier' of D;
  now
    let f be Morphism of C opp;
    thus *'S.f = S.(opp f) by Def10
      .= S.((*id C).f) by Th65
      .= (S*(*id C)).f by FUNCT_2:15;
  end;
  hence *'S = S*(*id C) by FUNCT_2:63;
  now
    let f be Morphism of C;
    thus S*'.f = (S.f) opp by Def11
      .= (id* D).(S.f) by Th63
      .= ((id* D)*S).f by FUNCT_2:15;
  end;
  hence thesis by FUNCT_2:63;
end;
