Ok, I will make an edit, I figured that if it was called a category that must be what it was. :-P

Do we know that \\( \mathcal{X} \\) will be a preorder?

Doesn't that depend on the monoidal preorder \\( \mathcal{V} \\) that we select.

Are you saying that any monoidal preorder \\( \mathcal{V} \\) will cause \\( \mathcal{X} \\) to be a preorder?

That seems likely.

How to prove that...

Do we know that \\( \mathcal{X} \\) will be a preorder?

Doesn't that depend on the monoidal preorder \\( \mathcal{V} \\) that we select.

Are you saying that any monoidal preorder \\( \mathcal{V} \\) will cause \\( \mathcal{X} \\) to be a preorder?

That seems likely.

How to prove that...