The category $\mathbf{Sup}$ of sup-lattices (posets admitting all supremum and supremum preserving map between them) is a well known example of a $*$-autonomous category:
The internal Hom is simply the set of supremum preserving map with the pointwise ordering between them, map $X \otimes Y \to Z$ corresponds to order preserving map $X \times Y \to Z$ that preserves supremum in each variable, the $*$-involution takes a poset $P$ to its opposite ordering and map $f:P \to Q$ to its right adjoint, seen as a supremum preserving map $Q^\text{op} \to P^\text{op}$. The dualizing object is $\Omega^\text{op}$, with $\Omega$ the subobject classifier (so $\{0,1\}$).
In particular, the tensor product can be described by the usual:
$$ X \otimes Y = \text{Hom}(X,Y^\text{op})^\text{op} $$
i.e. as the set of infimum preserving maps $X^\text{op} \to Y$.
The category $\mathbf{Sup}$ is the category of "locally presentable posets, i.e. locally presentable $(0,1)$-categories". But as soon as we move one step up the categorical ladder, to the (bi)category $\mathbf{Pres}$ of locally presentable categories and colimit preserving (equivalently, left adjoint) functors between them, we no longer have a $*$-autonomous category.
Still much of this structure is still present on $\mathbf{Pres}$: It is a monoidal closed category, where the internal hom is given by the category of all left adjoint functor $C \to D$. The tensor product was constructed by Bird (see section 5 of his thesis) following previous work of Kelly, and is such that colimit preserving functors $C \otimes D \to E$ are the same as functors $C \times D \to E$ preserving colimits in each separate variables.
And in fact, $\mathbf{Pres}$ still feel very much like a $*$-autonomous category, again with opposite category and taking right adjoint as the $*$-involution, in the sense that we still have formulas like:
$$ \text{Ladj}(C,\text{Set}^\text{op}) \simeq C^\text{op} $$ $$ C \otimes D \simeq \text{Ladj}(C,D^\text{op})^\text{op}$$
But it is not a $*$-autonomous category for the good reason that the opposite a locally presentable category is almost never locally presentable (it is if and only if it is an object of $\mathbf{Sup} actually).
My question is, can we modify $\mathbf{Pres}$ (by adding new objects) to make it a $*$-autonomous category in a nice way ?
I am relatively open about the sort of category (let's call it $\mathbf{A}$) I want, here is a list of the type of properties I would like to have.
$\mathbf{A}$ should be a $*$-autonomous category.
$\mathbf{A}$ should contains $\mathbf{Pres}$ as a monoidal full subcategory, closed under exponential.
But I still would like objects of $\mathbf{A}$ to be category in some sense: , and the formula above fits into a $*$-autonomous structure of $\mathbf{A}$:
$\mathbf{A}$ should have a forgetfull functor $U$ to the category of categories and left adjoint functor between them.
$U$ restricted to $\mathbf{Pres}$ should be the obvious functor.
$U(C^*)$ should be $C^\text{op}$ and if $F$ is a functor $U(F^*)$ should be a right adjoint of $U(F)$.
I don't think it is reasonable to expect that $U$ could be fully faithful, so I'm fine with $\mathbf{A}$ being a category of "category with structure", as long as $U$ is fully faithful on $\mathbf{Pres}$.
$\mathbf{Pres}$ is also known to have all small limits and colimits, I would like these to be preserved by the inclusion to $\mathbf{A}$. I don't think it is reasonable to expect that $\mathbf{A}$ will have all small limits and colimits as well, but the more it has the better.
Any negative results showing that this whish list, or some other reasonable wish list is too strong would also be greatly appreciated.
Precision: I'm aware of Mike Shulman recent preprint showing that any monoidal closed category can be fully faithfuly embeded in a $*$-autonomous category, with an embedding preserving tensor product and exponential (and even some colimits). In fact I had this question in mind for a long time, but I've thinking about it again recently because of this preprint.
So, assuming Mike's construction apply to $\mathbf{Pres}$, despite it not being a $1$-category nor locally small, that is definitely a good place to start. But I havn't really been able to parse what this construction would give in this case, nor if it comes with a forgetful functor to Cat. Also, I got the impression that this construction was maybe to general for this example ? Any comment on this would also be very welcome.
