> I'm confused by this. So far we've been talking about \$$\mathcal{V}\$$-enriched categories when \$$\mathcal{V}\$$ is a monoidal preorder. So, for example, there's a monoidal preorder \$$\mathbf{Bool}\$$, and that's what lets us talk about \$$\mathbf{Bool}\$$-enriched categories.

I am so sorry for the confusion, I have tried to clean up the question.

I edited the question to define a category **DiscretePosEnrich**. The idea behind this new category is that consists of \$$\mathcal{V}\$$-enriched categories for arbitrary \$$\mathcal{V}\$$. The only constraint is that \$$\mathcal{V}\$$ must have a discrete poset underlying its monoidal preorder.

I hope that clears this up. I can prove this tomorrow if nobody finds it interesting.