reserve A,B,C for Category,
  F,F1,F2,F3 for Functor of A,B,
  G for Functor of B, C;
reserve m,o for set;
reserve t for natural_transformation of F,F1,
  t1 for natural_transformation of F1,F2;
reserve a,b for Element of C;

theorem Th33:
  for A being discrete Category, a being Object of A holds Hom(a,a ) = { id a}
proof
  let A be discrete Category, a be Object of A;
  now
    let g be Morphism of a,a;
    id a in Hom(a,a) & g in Hom(a,a) by CAT_1:def 5;
   hence g = id a by ZFMISC_1:def 10;
  end;
  hence thesis by CAT_1:8;
 end;
