Thanks for the catch, Simon! I fixed my argument in my post.

> 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.

Now that you mention this, it is very clear what is going on.