reserve B,C,D for Category;

theorem
  for S being Function of the carrier' of C, the carrier' of D holds *'S
  *' = *'(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.f) opp by Def11
      .= (S.(opp f)) opp by Def10
      .= (S*').(opp f) by Def11
      .= *'(S*').f by Def10;
  end;
  hence thesis by FUNCT_2:63;
end;
