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.

> 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.