There are three notions of equipment in the literature: * [Wood's](References#wood82proarrowsi): 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](References#shulman08framed): 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](References#ckvw98change): 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'. * There is also the related notion of a [[nlab:connection on a double category]]. **Question**: Why are these all equivalent? 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 $CatProf$. So we need to know what a [[biprofunctor]] is, and to check that biprofunctors form at least a bicategory (if not an equipment) that has well-behaved Kleisli objects. The Grothendieck construction for a monad on a 1-category in $BiProf$ should then give a double category whose underlying span is a two-sided fibration.