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

>what is the endofunctor category \$$\mathbf{N}^\mathbf{N}\$$?

**Summary**: Objects in \$$\mathbf{N}^\mathbf{N}\$$ correspond to scalar multipliers \$$k \times \cdot : \mathbb{N} \to \mathbb{N}\$$, while morphisms act like scalar addition \$$c+ \cdot : \mathbb{N} \to \mathbb{N} \$$. There is no morphism from one functor to another in \$$\mathbf{N}^\mathbf{N}\$$.

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We have seen that functors \$$F: \mathbf{N} \to \mathbf{N}\$$ are just scalar multipliers.

From Wikipedia's definition of a [*Functor Category*](https://en.wikipedia.org/wiki/Functor_category), we have:

> In category theory, a branch of mathematics, the functors between two given categories form a category, where the objects are the *functors* and the morphisms are *natural transformations* between the functors.

Wikipedia gives the following definition of *natural transformations*:

> A natural transformation \$$\eta\$$ from functors \$$F: \mathbf{C} \to \mathbf{D}\$$ to \$$G: \mathbf{C} \to \mathbf{D}\$$ is a family of morphisms that satisfies two requirements.

> 1. The natural transformation must associate to every object \$$X\$$ in \$$\mathbf{C}\$$ a morphism \$$\eta_X : F(X) → G(X)\$$ between objects of \$$\mathbf{D}\$$. The morphism \$$\eta_X\$$ is called the component of \$$\eta\$$ at \$$X\$$.
> 2. Components must be such that for every morphism \$$f : X \to Y\$$ in \$$\mathbf{C}\$$ we have:
> $$\eta_{Y}\circ_{\mathbf{D}} F(f) =G(f)\circ_{\mathbf{D}} \eta _{X}$$

There is only one object in \$$\mathbf{N}\$$ and it is \$$\star\$$. The definition of natural transformation simplifies to:

$\eta _{\star} \circ\_{\mathbf{N}} F(f) =G(f) \circ\_{\mathbf{N}} \eta _{\star} \tag{✴}$

Since both \$$\mathbf{C}\$$ and \$$\mathbf{D}\$$ are \$$\mathbf{N}\$$ for functors in \$$\mathbf{N} \to \mathbf{N}\$$, we have \$$\circ\_{\mathbf{D}} = \circ\_{\mathbf{N}}\$$.

Note that \$$\circ\$$ in \$$\mathbf{N}\$$ reflects addition and \$$\eta _{\star}\$$ corresponds to some \$$c \in \mathbb{N}\$$.

Also note that \$$F \in \mathbf{N}^\mathbf{N} \$$ reflects \$$k \times \cdot: \mathbb{N} \to \mathbb{N}\$$ for some \$$k \in \mathbb{N}\$$ and \$$G \in \mathbf{N}^\mathbf{N} \$$ reflects \$$l \times \cdot: \mathbb{N} \to \mathbb{N}\$$ for some \$$l \in \mathbb{N}\$$.

This means (✴) is corresponds to, for all \$$n \in \mathbb{N}\$$:

$c + k \times n = l \times n + c$

However, we know that addition is commutative and all scalar additions \$$c + \cdot : \mathbb{N} \to \mathbb{N}\$$ are injective. So we may cancel \$$c\$$ on either side of (✴). Thus we have for all morphisms \$$f \in \mathbf{Mor}(\mathbf{N}) \$$:

$k \times n =l \times n$

Hence \$$k = l\$$, and thus \$$F = G\$$.

This means that every natural transformation \$$\eta : F \to F\$$ is an endomorphism in \$$\mathbf{N}^\mathbf{N}\$$.

We also have that every \$$c \in \mathbf{Mor}(\mathbf{N})\$$ gives rise to some endomorphism for a functor \$$F : \mathbf{N} \to \mathbf{N}\$$.

To see this, note that for every functor there is constant \$$k \in \mathbb{N} \$$ such that \$$F(f) \in \mathbf{N} \$$ reflects \$$k \times f \in \mathbb{N}\$$. Since addition commutes in arithmetic, for all \$$c \in \mathbb{N} \$$ we have:

$c + k \times f = k \times f + c$

which corresponds to:

$\eta \circ_{\mathbf{N}} F(f) = F(f) \circ_{\mathbf{N}} \eta$

For all \$$\eta \in \mathbf{Mor}(\mathbf{N}) \$$.