Finn Lawler empty 4 (Rev #1)

I want to define a notion of cartesian equipment, the idea being that a bicategory $M$ should be a cartesian bicategory if and only if the proarrow equipment $Map M \to M$ is a cartesian equipment.

This shouldn’t be hard to do. There are three notions of equipment in the literature:

• Wood’s: an identity-on-objects, locally fully faithful pseudofunctor $K \to M$, written $f \mapsto f_*$, where each $f_*$ has a right adjoint $f^*$ in $M$.

• Shulman’s: a framed bicategory, that is, a pseudo double category $M$ whose source and target functors $(s,t) \colon M_1 \to M_0 \times M_0$ form a bifibration.

• that of Carboni–Kelly–Verity–Wood: a (normal) pseudofunctor $M \colon K^{op} \times K \to Cat$, where $K$ is a 1-category.

The last is strictly more general than the others, as are even their starred pointed equipments, i.e. those equipped with a transformation $\hom_K \Rightarrow M$, where the left and right actions of $K$ have suitable adjoints. These CKVW equipments should be equivalent to the others if we ask for a transformation $M^2 \Rightarrow M$ making $M$ a (pseudo)monad in a suitable bi- or tricategory of ‘biprofunctors’.

Question: Why are these all equivalent?

With the correct notion of equipment, there are two possible notions of adjoint morphisms, and hence limits: those of CKVW, and those of Grandis–Paré (for double categories).

Question: Do these notions coincide?

Then a cartesian equipment will be an equipment that ‘has finite products’, that is an equipment $M$ for which the diagonals $M \to M^2$ and $M \to 1$ have right pseudoadjoints.