John wrote:

> 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}\$$.

You are right.

I attempted to clean this up above:

- The objects of this category are defined to be \$$\mathrm{Obj}(\mathbf{Mult}) = \lbrace\mathbf{Mor}(\mathbf{N})\rbrace\$$. In other words there is a single object.
- 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\$$

Since there's only one object, every morphism fixes it.

This category is the same as the monoid \$$\langle \mathbb{N}, \times, 1\rangle\$$ Jonathan mentioned in [#1](https://forum.azimuthproject.org/discussion/comment/19163/#Comment_19163).

> 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.

This wasn't what I had in mind but I am curious to know the name of this construction.