As an exceptionally silly case, what does a **Unit**-enriched category look like? **Unit** is the single-element set \$$\\{\cdot\\}\$$ with the trivial preorder -- the only one it could have, after all -- and the natural monoidal structure -- again, the only one it could possibly have.

For a given set \$$S\$$, we can define the **Unit**-enriched category \$$\mathcal{X}\$$ by \$$\mathrm{Ob}(\mathcal{X}) = S\$$ and \$$\mathcal{X}(x, y) = \cdot\$$ for all \$$x, y \in S\$$. This satisfies both conditions trivially.

I'm tempted to identify this construction with the codiscrete preorder on \$$S\$$.