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?