>**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)}. \$