On Sat, 29 Apr 2006, Vaughan Pratt wrote:
As an initial guess: "categories", with dualizer Set, which is both a category and a distributive category. So is CAT(C,Set) a distributive category? And if so, is DCAT(CAT(C,Set),Set) equivalent to C? And what about the enriched case V-DCAT(V-CAT(C,V),V)?
There's a result related to this (though not, I think, actually containing it) in my paper with Andre Joyal on continuous categories and exponentiable toposes (JPAA 25, 1982). We showed that there is an equivalence (which is really a duality, but the arrows on one side have been reversed) between quasi-injective toposes (that is, toposes which occur as retracts of presheaf toposes) and the categories that occur as their categories of points, which are continuous categories satisfying a certain size restriction. Peter Johnstone