Thanks Tom - and everyone else - this is really helpful. In fact my suppostion that P is a functor from C^op x D to Set was too special. P can be a functor from C^op x D to any category E. This more general version is the one I was told by Rasmus Mogelberg and is also (enriched) the one that appears in the Kelly-Lack paper. Accordingly we modify your explanation.
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)?
Your question now becomes: "What is the weighted limit {Q,P}?" BTW this is a good illustration of the conceptual benefit of weighted limits even in ordinary category theory. Paul
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).
-- Paul Blain Levy School of Computer Science, University of Birmingham +44 (0)121 414 4792 http://www.cs.bham.ac.uk/~pbl [For admin and other information see: http://www.mta.ca/~cat-dist/ ]