John, Matthew: I think there is a mild problem with your construction of a non-trivial functor \$$F\colon \mathbf{Set}\to \mathbb{N}\$$. You have the map \$$f\colon \emptyset\to \mathbb{R}\$$, with \$$F(f)=n\$$, and with all other functions going to \$$\mathrm{id}\$$. Let \$$t \colon \mathbb{R}\to \\{\ast\\}\$$ be the function that send every number to \$$\ast\$$. Of course with that domain and that codomain you have no choice. Then we have the composite \$$t\circ f\colon \emptyset \to \(\ast$$\\). We should then have \$$n=\mathrm{id}\circ n = F(t)\circ F(f)= F(t\circ f) =\mathrm{id} \$$.

We need to send *all* initial functions to \$$n\$$. I guess we have a functor from \$$\mathbf{Set}\$$ to the category with two objects \$$a\$$ and \$$b\$$ and with precisely one non-identity morphism \$$a\to b\$$: the empty set gets sent to \$$a\$$, everything else goes to \$$b\$$. The functor you want from \$$\mathbf{Set}\$$ to \$$\mathbb{N}\$$ factors through this.