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 case M = 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.

A complete lattice L is constructively completely distributive, (CCD), when the sup arrow from down-closed subobjects of L to L has a left adjoint. The Karoubian envelope of the bicategory of relations is biequivalent to the bicategory of (CCD) lattices and sup-preserving arrows. There is a restriction to order ideals and "totally algebraic" lattices. Both biequivalences have left exact versions. As applications we characterize projective sup lattices and recover a known characterization of projective frames. Also, the known characterization of nuclear sup lattices in set as completely distributive lattices is extended to yet another characterization of (CCD) lattices in a topos. Research partially supported by grants from NSERC Canada. Diagrams typeset using Michael Barr's diagram package. AMS Subject Classification Primary: 06D10 Secondary 18B35, 03G10. Keywords: completely distributive, adjunction, projective, nuclear Introduction Idempotents do not split in the category of rel...

A complete lattice L is constructively completely distributive, (CCD)(L), if the sup map defined on down closed subobjects has a left adjoint. We characterize preservation of this property by left exact functors between toposes using a "logical comparison transformation". The characterization is applied to (direct images of) geometric morphisms to show that local homeomorphisms (in particular, product functors) preserve (CCD) objects, while preserving (CCD) objects implies openness. Research partially supported by grants from NSERC Canada. Diagrams typeset using Michael Barr's diagram macros. AMS Subject Classifications: 18B35, 06D10, 03G10. Introduction A complete ordered set L is constructively completely distributive, abbreviated to (CCD)(L); if there is a left adjoint to the sup map W : DL \Gamma! L; where DL is the set of down-closed subobjects of L ordered by inclusion. The condition is equivalent to (8S ` DL) i n S j S 2 S o = n fT (S) j S 2 Sg j T 2 \PiS oj which...