Regarding [comment #4](

> 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](
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]( that there exist preorders that cannot be given monoidal structure (for example, [Matthew's]( and [Anindya's]( examples).