> You haven't said this outright, but the morphisms of this category are functors, correct?

Yes. The morphisms are the functors \\(F : \mathbf{N} \to \mathbf{N}\\)

I went back and fixed my post to make this more explicit.

> The functors we've been discussing are of kind \\(\mathbf{N} \to \mathbf{N}\\). How do these associate to morphisms in \\(\mathbf{Mult}\\)?

Each functor corresponds to a morphism and vice versa.