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.

> 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.