> **Puzzle 121.** Let \$$\mathbf{N}\$$ be the free category on this graph:
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>
![N](http://math.ucr.edu/home/baez/mathematical/7_sketches/graph_loop.png)

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> It has one morphism for each natural number. What are all the functors \$$F: \mathbf{N} \to \mathbf{N}\$$?

As Jonathan Castello writes \$$\langle \mathbf{N}, \circ, id_{\star} \rangle \$$ isomorphic to the monoid \$$\langle \mathbb{N}, +, 0\rangle\$$.

An endofunctor \$$F\$$ obeys the law:

$F(x \circ y) = F(x) \circ F(y)$

Hence \$$F\$$ is a bit like a [linear map](https://en.wikipedia.org/wiki/Linear_map). We also have

$F(id_{\star}) = id_{\star}$

So we know that \$$F\$$ is kind of [*scaling*](https://en.wikipedia.org/wiki/Scaling_(geometry)).

Hence the functors \$$F\$$ behave like multiplying constants on \$$\mathbb{N}\$$. There is one for each morphism in \$$\mathbb{N}\$$.

> What happens when we compose them?

For one, we get a category, which I am going to call \$$\mathbf{Mor}\$$

- 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\$$

As Jonathan alluded to in [#1](https://forum.azimuthproject.org/discussion/comment/19163/#Comment_19163), this category is like the covariant [Cayley representation](https://en.wikipedia.org/wiki/Cayley%27s_theorem#Proof_of_the_theorem) of \$$\langle \mathbb{N}, \times, 1\rangle\$$.

Now that we have \$$\mathbf{Mult}\$$, I have another idea...

**Puzzle MD 1**: Consider the functors \$$E : \mathbf{N} \to \mathbf{Mult}\$$. What are these functors like?