Matthew wrote:

> For one, we get a category, which I am going to call \\(\mathbf{Mult}\\)

> - The objects are the morphisms of \\(\mathbf{N}\\)
> - The identity is the identity functor \\(\mathbf{1}_{\bullet}\\), corresponding to the scaling \\(1 \times \cdot\\)
> - Morphisms in this category are functors \\(F : \mathbf{N} \to \mathbf{N}\\)
> - Morphism composition is functor composition \\(\bullet\\)

In a category, each morphism goes from one particular object to one particular object. You're not specifying how this works for \\(\mathbf{Mult}\\).

But I can guess: morphisms in \\(\mathbf{Mult}\\) are not really functors \\(F : \mathbf{N} \to \mathbf{N}\\); rather, each functor \\(F : \mathbf{N} \to \mathbf{N}\\) gives infinitely many morphisms in \\(\mathbf{Mult}\\), one for each object of \\(\mathbf{Mult}\\). Objects of \\(\mathbf{Mult}\\) can be identified with natural numbers, and if a functor \\(F : \mathbf{N} \to \mathbf{N}\\) maps the object \\(n\\) to the object \\(n'\\), we decree that there's a morphism

\[ (F,n) : n \to n' \]

in \\(\mathbf{Mult}\\). We compose these in the obvious way:

\[ (F',n') \circ (F,n) = (F' \circ F, n) .\]

This is an example of a sort of well-known construction.