I want to define a notion of **cartesian equipment**, the idea being that a [[nlab:bicategory]] $M$ should be a [[nlab:cartesian bicategory]] if and only if the [[nlab: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? +-- {: .standout} An identity-on-objects (pseudo)functor that is locally fully faithful is essentially the same thing as an identity-on-objects functor out of a locally discrete bicategory. In the case of strict 2-categories, these are (by some enriched-category nonsense) precisely the Kleisli objects for monads (on locally discrete strict 2-categories) in $Cat{-}Prof$. The Grothendieck construction for such a profunctor should give a double category whose underlying span is a two-sided fibration. =-- 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.