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.