reserve C for Category,
  C1,C2 for Subcategory of C;

theorem Th32:
  for C being Category, o being Object of C, f being Element of o Hom
  for a being Object of o-SliceCat C st a = f holds id a = [[a,a], id cod f]
proof
  let C be Category, o be Object of C, f be Element of o Hom;
  let a be Object of o-SliceCat C;
  assume
A1: a = f;
  consider b,c being Element of o Hom, g being Morphism of C such that
A2: id a = [[b,c], g] and
A3: dom g = cod b and g(*)b = c by Def12;
A4: dom id cod f = cod f;
  f = (id cod f)(*)f by CAT_1:21;
  then reconsider h = [[f,f], id cod f] as Morphism of o-SliceCat C by A4,Def12
;
A5: (id a)`11 = b by A2,MCART_1:85;
A6: (id a)`12 = c by A2,MCART_1:85;
A7: dom id a = b by A5,Th2;
A8: cod id a = c by A6,Th2;
A9: b = a by A7;
A10: c = a by A8;
  cod h = h`12 by Th2
    .= a by A1,MCART_1:85;
  then h = (id a)(*)h by CAT_1:21
    .= [[f,f], g(*)id cod f] by A1,A2,A9,A10,Def12
    .= [[f,f], g] by A1,A3,A9,CAT_1:22;
  hence thesis by A1,A2,A8,A9;
end;
