There is a one-to-one correspondence between preorders and Bool-categories
Construct preorder \((X,\leq_X)\) from any Bool-category \(\mathcal{X}\)
Let \(X\) be \(Ob(\mathcal{X})\) and \(x\ \leq_X\ y\) be defined as \(\mathcal{X}(x,y)=true\)
This is reflexive and transitive because of the two constraints we put on enriched categories.
The first constraint says that \(true \leq (x \leq_X x)\)
The second constraint says \((x \leq_X y) \land (y \leq_X z) \leq (x \leq_X z)\)
Construct Bool-category from a preorder:
Let \(Ob(X)=X\) and define \(\mathcal{X}(x,y)=true\) iff \(x \leq_X y\)
The two constraints on preorders give us our two required constraints on enriched categories.