A pointed unary system is a tuple $(A, q, f)$ consisting of a set $A$ with a point $q\in A$ and a unary function $f: A\rightarrow A$, and the natural numbers $(\mathbb{N}, 0, s)$ are defined as the initial object in the category of pointed unary systems.
Now, since I am not aware of an official term for the following object, suppose we define a pointed unary system $(A, q, f)$ where $f$ is a bijective function to be a 'pointed invertible unary system'. Is the initial object of the category of pointed invertible unary systems the integers $(\mathbb{Z}, 0, s)$?
Sure: assuming your morphisms are functions are morphisms respecting $q$ and $f$, then such an object is simply a set equipped with a $\mathbb{Z}$-action and a chosen point. The initial such object is certainly the free set with a $\mathbb{Z}$-action on one generator, with is $\mathbb{Z}$ itself.