reserve B,C,D for Category;

theorem
  for D being strict Category, S being Function of the carrier' of C,
  the carrier' of D holds S*'*' = S
proof
  let D be strict Category;
  let S be Function of the carrier' of C, the carrier' of D;
  now
    thus D opp opp = D by FUNCT_4:53;
    let f be Morphism of C;
    thus S*'*'.f = (S*'.f) opp by Def11
      .= (S.f) opp opp by Def11
      .= S.f;
  end;
  hence thesis by FUNCT_2:63;
end;
