Matthew wrote:

> Hopefully someone can come along and say definitively what the objects are in the category of endofunctors of a category \\(\mathcal{C}\\).

The objects of the endofunctor category of \\(\mathcal{C}\\) are functors \\(F : \mathcal{C} \to \mathcal{C}\\). The morphisms are natural transformations between such functors.

This is a special case of something we'll talk about soon. Given two categories \\(\mathcal{C}, \mathcal{D}\\), the **functor category** \\(\mathcal{D}^\mathcal{C}\\) has:

* functors \\(F: \mathcal{C} \to \mathcal{D}\\) as objects,

* natural transformations between such functors as morphisms.

In this notation the endofunctor category of \\(\mathcal{C}\\) is \\(\mathcal{C}^\mathcal{C}\\).

We haven't talked about natural transformations in this course yet, so the following puzzles are "unfair", but I can't resist mentioning them. Maybe I'll make them official puzzles later.

**Puzzle.** If \\(\mathbf{N}\\) is the free category on the graph

what is the endofunctor category \\(\mathbf{N}^\mathbf{N}\\)?

**Puzzle.** What is the functor category

\[ \mathbf{N}^{(\mathbf{N}^\mathbf{N})} ? \]

We could go on...

> Hopefully someone can come along and say definitively what the objects are in the category of endofunctors of a category \\(\mathcal{C}\\).

The objects of the endofunctor category of \\(\mathcal{C}\\) are functors \\(F : \mathcal{C} \to \mathcal{C}\\). The morphisms are natural transformations between such functors.

This is a special case of something we'll talk about soon. Given two categories \\(\mathcal{C}, \mathcal{D}\\), the **functor category** \\(\mathcal{D}^\mathcal{C}\\) has:

* functors \\(F: \mathcal{C} \to \mathcal{D}\\) as objects,

* natural transformations between such functors as morphisms.

In this notation the endofunctor category of \\(\mathcal{C}\\) is \\(\mathcal{C}^\mathcal{C}\\).

We haven't talked about natural transformations in this course yet, so the following puzzles are "unfair", but I can't resist mentioning them. Maybe I'll make them official puzzles later.

**Puzzle.** If \\(\mathbf{N}\\) is the free category on the graph

what is the endofunctor category \\(\mathbf{N}^\mathbf{N}\\)?

**Puzzle.** What is the functor category

\[ \mathbf{N}^{(\mathbf{N}^\mathbf{N})} ? \]

We could go on...