Jonathan

So my question was actually

>According to the definition above, \$$x,y \in \mathrm{Ob}(\mathcal{X})\$$. But in order for \$$\mathcal{X}(x,y)\$$ to be an element of \$$\mathcal{V}\$$, \$$x,y \in V\$$ must be true also? Then what is the difference between \$$\mathrm{Ob}(\mathcal{X})\$$ and \$$V\$$?

And you showed me that the answer is no: the set \$$V\$$ is the set which you enrich the arrows of the category. So there are two layers to this configuration, object and arrows of the category and the monoidal preorder which decorates the arrows.

I think I got confused because I thought both sets were part of a preorder but one was actually the objects of a category which brings me to another question.

Can we enrich a preorder? To me its seems the category itself is also a preorder or can be made into a preorder since we are trying to answer the quantified version of the same question. Why do we have to define it as category instead of a preorder?