reserve U for Universe;
reserve x for Element of U;
reserve U1,U2 for Universe;

theorem
  for C being U-locally_small Category st the carrier of C is U-set holds
  C is U-small
  proof
    let C be U-locally_small Category;
    assume
A1: the carrier of C is U-set;
    then union the set of all Hom(a,b) where
       a,b is Object of C is Element of U &
       the carrier' of C c= union the set of all Hom(a,b)
       where a,b is Object of C by CAT_1:92,Th98;
    hence thesis by A1,CLASSES4:13;
  end;
