Natural Numbers Object Definition

An object which has a distinguished global element (which may be called z, for “zero") and a distinguished endomorphism (which may be called s, for “successor") such that iterated compositions of s upon z (i.e., s^n \circ z) yields other global elements of the same object which correspond to the natural numbers (s^n \circ z \leftrightarrow n). Such object has the universal property that for any other object with a distinguished global element (call it z') and a distinguished endomorphism (call it s'), there is a unique morphism (call it φ) from the given object to the other object which maps z to z' (\phi \circ z = z') and which commutes with s; i.e., \phi \circ s = s' \circ \phi.