[Matthew](https://forum.azimuthproject.org/discussion/comment/18420/#Comment_18420), observe that \$$\mathcal{X}(\cdot, \cdot)\$$ effectively gives a function \$$\mathcal{X} : \mathrm{Ob}(\mathcal{X}) \times \mathrm{Ob}(\mathcal{X}) \to \mathcal{V}\$$. Therefore, the two properties on \$$\mathcal{V}\$$-enriched categories are statements about how objects in \$$\mathcal{V}\$$ interact, not how objects in \$$\mathrm{Ob}(\mathcal{X})\$$ interact. We do not have any obvious way of producing a monoidal structure on \$$\mathcal{X}\$$.

(I would argue that, at least at this stage, a \$$\mathcal{V}\$$-enriched category is little more than a certain kind of function \$$\mathcal{X} : S \times S \to \mathcal{V}\$$ from a product of sets to a monoidal preorder, since we can always recover \$$\mathrm{Ob}(\mathcal{X})\$$ from the diagonal elements \$$(x, x) \in \mathrm{Dom}(\mathcal{X})\$$.)