categories: Enriched category theory, again

[Mark Hovey <hovey@picard.math.wesleyan.edu>]

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