> 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.

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.