Regarding [comment #4](https://forum.azimuthproject.org/discussion/comment/18377/#Comment_18377):

> Matthew Doty - in Puzzle 87 I was implicitly making **Bool** into a symmetric monoidal poset as in Puzzle 86. With this choice, it's not true that **Bool**-enriched categories are symmetric monoidal preorders.

>

> What are they really?

My take is that \\(\mathbf{Bool}\\)-enriched categories are preorders, but they are not monoidal.

\\(\mathbf{Bool}\\)-enriched categories are preorders, because we can construct a preorder from a \\(\mathbf{Bool}\\)-enriched category, and viceversa;

and the two constructions are inverses of each other (we have to prove this in [exercise 2.31](https://forum.azimuthproject.org/discussion/1988)).

For example, we can obtain a preorder \\((X, \le_X)\\) from a \\(\mathbf{Bool}\\)-enriched category \\(\mathcal{X}\\) as follows:

- \\( X = \mathrm{Ob}(\mathcal{X}) \\)

- \\( x \le_X y \\) iff. \\( \mathcal{X}(x, y) = \tt{true} \\).

However, I don't think \\(\mathbf{Bool}\\)-enriched categories are necessarily monoidal:

we have seen in puzzle 66 from [lecture 21](https://forum.azimuthproject.org/discussion/2082/lecture-21-chapter-2-monoidal-preorders/p1) that there exist preorders that cannot be given monoidal structure (for example, [Matthew's](https://forum.azimuthproject.org/discussion/comment/17926/#Comment_17926) and [Anindya's](https://forum.azimuthproject.org/discussion/comment/17927/#Comment_17927) examples).

> Matthew Doty - in Puzzle 87 I was implicitly making **Bool** into a symmetric monoidal poset as in Puzzle 86. With this choice, it's not true that **Bool**-enriched categories are symmetric monoidal preorders.

>

> What are they really?

My take is that \\(\mathbf{Bool}\\)-enriched categories are preorders, but they are not monoidal.

\\(\mathbf{Bool}\\)-enriched categories are preorders, because we can construct a preorder from a \\(\mathbf{Bool}\\)-enriched category, and viceversa;

and the two constructions are inverses of each other (we have to prove this in [exercise 2.31](https://forum.azimuthproject.org/discussion/1988)).

For example, we can obtain a preorder \\((X, \le_X)\\) from a \\(\mathbf{Bool}\\)-enriched category \\(\mathcal{X}\\) as follows:

- \\( X = \mathrm{Ob}(\mathcal{X}) \\)

- \\( x \le_X y \\) iff. \\( \mathcal{X}(x, y) = \tt{true} \\).

However, I don't think \\(\mathbf{Bool}\\)-enriched categories are necessarily monoidal:

we have seen in puzzle 66 from [lecture 21](https://forum.azimuthproject.org/discussion/2082/lecture-21-chapter-2-monoidal-preorders/p1) that there exist preorders that cannot be given monoidal structure (for example, [Matthew's](https://forum.azimuthproject.org/discussion/comment/17926/#Comment_17926) and [Anindya's](https://forum.azimuthproject.org/discussion/comment/17927/#Comment_17927) examples).