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 Th35:
  for A being discrete Category, B being Category, F being Functor
  of B,A, t being transformation of F,F holds t = id F
proof
  let A be discrete Category, B be Category, F be Functor of B,A, t be
  transformation of F,F;
  now
    let a be Object of B;
A1: Hom(F.a,F.a) = { id(F.a) } by Th33;
    t.a in Hom(F.a, F.a) by CAT_1:def 5;
    hence t.a = id(F.a) by A1,TARSKI:def 1
      .= (id F).a by Th16;
  end;
  hence thesis by Th15;
end;
