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).