I am back again, with two more questions about enriched category theory. As usual, what I am really searching for are references so I can avoid writing any category-theoretic proofs myself. The good news is that Max Kelly's book on enriched category theory is actually making sense to me now (thank you, Max!). I think these two questions are not answered in there, but I could be wrong. The first is a generalization of the enriched Yoneda lemma. Recall that this says that [A,V](X^*,F) is isomorphic to FX. Here [A,V] denotes the V-category of V-functors from A to V, X is an object of A, and X^* is the V-functor that takes Y to A(X,Y). (and A is small). What I want is [A,D](Y tensor X^*,F) is isomorphic to D(Y,FX). Here D is a V-category that is tensored over V, X is an object of A, and Y is an object of D. The functor Y tensor X^* takes Z to Y tensor A(X,Z). If you take D = V and Y = the unit of V, you recover the Yoneda lemma above. I don't remember if any completeness or cocompleteness assumptions on D are necessary here because I always assume D is bicomplete. Has this stonger Yoneda lemma appeared in print anywhere? I think the proof is the same as the usual Yoneda lemma. My second question has to do with Brian Day's old work. He shows that if A is a small symmetric monoidal V-category, then [A,V] is a closed symmetric monoidal V-category. What I want to say is that if D is a (bicomplete) V-category that is tensored and cotensored over V, then [A,D] is an [A,V]-category that is tensored and cotensored over [A,V] (A is as above, a small symmetric monoidal V-category). This is obviously a generalization of Day's work, and is obviously proved by following along in Day and changing a few V's to D's. Has it ever appeared in print? What I am actually doing (with Manos Lydakis), in case anyone is wondering why I am asking all these questions, is examining the homotopy theory of [A,D]. So I assume V and D are themselves model categories and put different model structures on [A,D]. For certain A you recover what algebraic topologists call spectra and symmetric spectra. The main goal is to explain and generalize the Goodwillie calculus of functors. Mark Hovey 12-Jan-2002 09:09:20 -0400,2877;000000000000-00000000