> **Puzzle 159.** The categories I've been calling \\(\mathcal{C}\\) and \\(\mathcal{D}\\) have other, more purely mathematical names. More precisely, they are _isomorphic_ to two other categories we've already seen in this course, which have more mathematical names. What are those other categories?

Matthew wrote:

> I think \\(\mathcal{C}\\) has been called "\\(\mathbf{2}\\)" in other discussions and \\(\mathcal{D}\\) has been called "\\(\mathbf{1}\\)".

That's right! They're part of a sequence of important categories, which are actually posets:









and so on.

Jade wrote:

> Puzzle 159: Is it the arrow category and the terminal category respectively?

That's also write! In previous puzzles we saw that:

* A functor \\(F: \mathbf{1} \to \mathcal{A}\\) is the same as an object of the category \\(\mathcal{A}\\).

* A functor \\(F: \mathbf{2} \to \mathcal{A}\\) is the same as an arrow, or morphism, in the category \\(\mathcal{A}\\),

* There is exactly one functor \\(F: \mathcal{A} \to \mathbf{1} \\). Thus, we call \\(\mathbf{1}\\) the **terminal** category.

* There is exactly one functor \\(F :\mathbf{0} \to \mathcal{A} \\). Thus, we call \\(\mathbf{0}\\) the **initial** category.

I didn't introduce the terms 'initial' and 'terminal', but we should!