Now for fixed , (u1) and (u2) form indeed a universal property, i.e. if such a function exists then it's unique and is fully caracterized by these properties. Unicity is proven by means of induction: assume there's functions and which satisfy (u1) and (u2) but which disagree i.e. on some list . Now by (u2) they also need to disagree on , hence must have length zero contradicting (u1).