A 2-categorical approach to change of base and geometric morphisms II

We introduce a notion of equipment which generalizes the earlier notion of pro-arrow equipment and includes such familiar constructs as $\rel\K$, $\spn\K$, $\par\K$, and $\pro\K$ for a suitable category $ \K$, along with related constructs such as the $\V$-$\pro$ arising from a suitable monoidal category $\V $. We further exhibit the equipments as the objects of a 2-category, in such a way that arbitrary functors $F:\eL -> \K$ induce equipment arrows $\rel F:\rel\eL ->\rel\K$, $\spn F:\spn\eL -> \spn\K$, and so on, and similarly for arbitrary monoidal functors $\V -> \W$. The article I with the title above dealt with those equipments $\M$ having each $\M(A,B)$ only an ordered set, and contained a detailed analysis of the case $\M =\rel\K$; in the present article we allow the $\M(A,B)$ to be general categories, and illustrate our results by a detailed study of the case $\M=\spn\K$. We show in particular that $\spn$ is a locally-fully-faithful 2-functor to the 2-category of equipments, and determine its image on arrows. After analyzing the nature of adjunctions in the 2-category of equipments, we are able to give a simple characterization of those $\spn G$ which arise from a geometric morphism $G$.