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]] --- an equipment is a (pseudo) double category with both a connection and an 'op-connection', the latter giving e.g. the right adjoints $f^*$ in Wood's definition. **Question**: Why are these all equivalent? Let's forget about the 'op-connections' for the moment. A pseudofunctor $K \to M$ that is the identity on objects and locally fully faithful is the same as an identity-on-objects pseudofunctor $K' \to M$ out of the locally discrete bicategory given by simply throwing away the 2-cells of $K$. An identity-on-objects pseudofunctor $K \to M$ exhibits $M$ as the Kleisli object of a monad on $K$ in the 2-category of [[biprofunctors]] (q.v.). This will be the associated CKVW equipment. ...