John wrote:

> For these two reasons it's hopeless (I believe) to develop the theory of enriched profunctors when \$$\mathcal{V}\$$ is merely monoidal. Thus, I could have spared myself some grief by assuming \$$\mathcal{V}\$$ is a commutative monoidal poset throughout this lecture. But in fact closed monoidal categories, and closed monoidal posets, are interesting even in the noncommutative case! So I couldn't resist getting into trouble.

Okay, that makes things clearer.

> The reason is that in the noncommutative case one must really distinguish between **left closed** and **right closed** monoidal posets. In one, the operation \$$a \otimes -\$$ has a right adjoint for each \$$a\$$. In the other, the operation \$$- \otimes a\$$ has a right adjoint for each \$$a\$$.
>
> (When we want to distinguish these two, we do it by turning around the lollipop in \$$\multimap\$$. I don't know how to do that in this TeX installation!)

I think the usual symbol is \multimapinv, but it doesn't work here. However, if you don't mind copy-pasting unicode, you can use \$$⟜\$$ which works fine.