Matthew Doty wrote:
> For one, we get a category, which I am going to call \$$\mathbf{Mult}\$$

You haven't said this outright, but the morphisms of this category are functors, correct? The functors we've been discussing are of kind \$$\mathbf{N} \to \mathbf{N}\$$. How do these associate to morphisms in \$$\mathbf{Mult}\$$?