reserve x, y for set;
reserve D for non empty set;
reserve UN for Universe;
reserve C for Category;
reserve O for non empty Subset of the carrier of C;
reserve G,H for AddGroup;
reserve V for Group_DOMAIN;

theorem
  for a being Object of GroupCat(UN), aa being Element of
  GroupObjects(UN) st a = aa holds id a = ID aa
 proof let a be Object of GroupCat UN, aa be Element of GroupObjects UN;
  set C = GroupCat UN;
  assume
A1:  a = aa;
   reconsider ii = ID aa as Morphism of C;
   reconsider ia = ii as GroupMorphismStr;
A2: dom ii = dom ia by Def25
      .= a by A1;
   cod ii = cod ia by Def26
      .= a by A1;
   then reconsider ii as Morphism of a,a by A2,CAT_1:4;
  for b being Object of C holds
     (Hom(a,b) <> {} implies
       for f being Morphism of a,b holds f(*)ii = f)
   & (Hom(b,a) <> {} implies
     for f being Morphism of b,a holds ii(*)f = f) by A1,Lm2;
  hence id a = ID aa by CAT_1:def 12;
 end;
