From this lecture:

>So, we get a truth value for each pair \\(x,y\\). Let's make up a relation \\(\le\\) on \\( \mathrm{Ob}(\mathcal{X})\\) such that \\(x \le y\\) is true when \\( \mathcal{X}(x,y) = \texttt{true}\\), and false when \\( \mathcal{X}(x,y) = \texttt{false}\\).

I think not knowing that we have to define another relation \\(\le\\) on \\( \mathrm{Ob}(\mathcal{X})\\) is what confused me the most when I first attempted to understand the past few lectures. Because there was nothing in the definition of enriched categories on this new relation, the relation \\(\le\\) from the monoidal preorder \\(\mathcal{V}\\) just seemed to "spill over" into \\(\mathrm{Ob}(\mathcal{X})\\) which made it unclear for the beginner which set was enriching what and which elements from the preorder was doing the enriching.

So from the definition of **Bool**-categories, you get a set of objects and a truth value in between them. My question is wouldn't you need to define a new relation like the statement above to make **Bool**-category into a preorder? Or is there some unspoken thing going on where you don't need this?

>So, we get a truth value for each pair \\(x,y\\). Let's make up a relation \\(\le\\) on \\( \mathrm{Ob}(\mathcal{X})\\) such that \\(x \le y\\) is true when \\( \mathcal{X}(x,y) = \texttt{true}\\), and false when \\( \mathcal{X}(x,y) = \texttt{false}\\).

I think not knowing that we have to define another relation \\(\le\\) on \\( \mathrm{Ob}(\mathcal{X})\\) is what confused me the most when I first attempted to understand the past few lectures. Because there was nothing in the definition of enriched categories on this new relation, the relation \\(\le\\) from the monoidal preorder \\(\mathcal{V}\\) just seemed to "spill over" into \\(\mathrm{Ob}(\mathcal{X})\\) which made it unclear for the beginner which set was enriching what and which elements from the preorder was doing the enriching.

So from the definition of **Bool**-categories, you get a set of objects and a truth value in between them. My question is wouldn't you need to define a new relation like the statement above to make **Bool**-category into a preorder? Or is there some unspoken thing going on where you don't need this?