Is the NNO in Cat isomorphic to $\mathbb {N}$ viewed as a discrete, skeletal category?

Question

Suppose N is any category which satisfies following axioms:

1. There exists a distinguished object, z.
2. There exists a distinguished functor, $\sigma': N\rightarrow N$.
3. To any category $X$ with a distinguished object $\bar{x}$ and a functor $F:X\rightarrow X$, there exists a unique (I know, many consider it evil!) functor $F_\bar{x}:N\rightarrow X$ which respects the structure $\sigma, F$ and maps z to $\bar{x}$.

We define Peano category as category $\mathbb{N}$ with following structure:

1. $\mathbb{N}$ is discrete skeleton category.
2. There exist a distinguished object, 0.
3. There exists a object-injective functor $\sigma: \mathbb{N}\hookrightarrow \mathbb{N}$.
4. 0 is not in the image of $\sigma$.
5. Induction: Let $B$ be a discrete-skeletal category with exactly two objects $(t, f)$. Let $\phi:\mathbb{N}\rightarrow B$ be any map. If $\phi(0) = true$ and $\phi(n) \Rightarrow \phi(\sigma(n))$, then every $n$ is mapped to true.

Based upon the above definitions, one can make following observations:

• N is infinite as $F_0$ is surjective.
• There is no arrow b/w any two distinct fibres defined by $F_0$.

Now, what I don't know is if (Question) every fibre contains exactly one object. If each fibre contains exactly one object, then inverse for $F_0$ can be constructed easily.

Any hint is greatly appreciated. Thank you in advance for your time.