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...