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...