Lax Presheaves and Exponentiability

The category of Set-valued presheaves on a small category B is a topos. Replacing Set by a bicategory S whose objects are sets and morphisms are spans, relations, or partial maps, we consider a category Lax(B,S) of S-valued lax functors on B. When S = Span, the resulting category is equivalent to Cat/B, and hence, is rarely even cartesian closed. Restricting this equivalence gives rise to exponentiability characterizations for Lax(B,Rel) in [9] and for Lax(B,Par) in this paper. Along the way, we obtain a characterization of those B for which the category UFL/B is a coreflective subcategory of Cat/B, and hence, a topos.