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 relK, spnK, parK, and proK 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 : L ✲K induce equipment arrows relF : relL ✲ relK, spnF : spnL ✲ spnK, 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 = relK; in the present article we allow the M(A,B) to be general categories, and illustrate our results by a detailed study of the caseM = spnK. 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 spnG which arise from a geometric morphism G.