In the comment #43 we had

>> It seems that Matthew is saying that it is not necessary that a monoidal preorder has the monoidal product distributing over joins for it to be closed.
>
> No, I didn't mean that.

Sorry, I should have quoted what I was referring to. In comment #21 you had

> **Lemma.** If \$$\mathcal{V}\$$ is monoidal partial order which also an [upper semilattice](https://en.wikipedia.org/wiki/Semilattice#Complete_semilattices) with arbitrary joins, then it is closed.

And afterwards you said

> The above argument doesn't need \$$\otimes\$$ to distribute over \$$\bigvee\$$ as far as I can tell. I believe it is necessary to define profunctor composition, however.