reserve B,C,D for Category;

theorem
  for C being strict Category, S being Function of the carrier' of C,
  the carrier' of D holds *'*'S = S
proof
  let C be strict Category;
  let S be Function of the carrier' of C, the carrier' of D;
  now
    thus C opp opp = C by FUNCT_4:53;
    let f be Morphism of C opp opp;
    thus *'*'S.f = *'S.(opp f) by Def10
      .= S.(opp opp f) by Def10
      .= S.f;
  end;
  hence thesis by FUNCT_2:63;
end;
