Finn Lawler empty 4 (Rev #3, changes)

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.

Outline

• See equipment? first.

With the correct notion of equipment, there are two possible notions of adjoint morphisms, and hence limits: those of Carboni–Kelly–Verity–Wood, 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.