>**Conjecture.** Any functor \\( F: \mathbf{Set} \to \mathbf{N}\\) must send every morphism in \\(\textbf{Set}\\) to the identity morphism.

>Can someone prove it or find a counterexample?

Would something like thing work?

\\[
F(X) := \begin{cases}
Id_{\mathbb{N}} & \text{ if } X =\varnothing, \\\\
succ \ \circ \ F(X \setminus \lbrace x \rbrace) & \text{ if } X \not= \varnothing,
\end{cases}
\\]

and,

\\[
F(\mathbf{Set}(X,Y)) := F(Y)^{F(X)}.
\\]