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}\\)?