Dear Paul, The following answer is essentially the same as Richard's, Ross's, and possibly Steve's (whose paper with Max I haven't looked up). But it provides a slightly different way of looking at things, which might be helpful if you haven't seen it already. Any adjunction F -| U: D --> C gives rise to a (D, C)-bimodule Q, that is, a functor Q: C^op x D --> Set. (Read "profunctor" or "distributor" if you prefer.) It is defined by Q(c, d) = D(Fc, d) = C(c, Ud). Your initial data gives such a (D, C)-bimodule Q, plus another (D, C)-bimodule P. Given such, we can ask: What is Hom(Q, P)? Calculating it one way, it's Hom(Q, P) = \int_{c, d} [Q(c, d), P(c, d)] = \int_{c, d} [D(Fc, d), P(c, d)] = \int_c P(c, Fc) by Yoneda. This used the definition of Q(c, d) as D(Fc, d). But we could also use the definition of Q(c, d) as C(c, Ud), and then a very similar calculation shows that Hom(Q, P) = \int_d P(Ud, d). So both your sets are canonically isomorphic to Hom(Q, P). Best wishes, Tom
Given categories C and D, a functor P : C^op x D --> Set and an adjunction F -| U : D --> C
the end over c in C of P(c,Fc) is (isomorphic to) the end over d in D of P(Ud,d).
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]