This is in reply to the queries in Mark Hovey's letter of 12 Jan, of which I shall try to include a copy below; I am limited by the inadequacies of this home computer. The first isomorphism he asks about, namely (using o for tensor product) [A,D](Y o A(X,-),F] =~ D(Y,FX), may be seen as a simple consequence of the fact that Y o A(X,-) is the left Kan extension of Y : I --> D along X : I --> A, where I is the unit V-category; see formula (4.18) or (4.25) of my book. The second question concerns an easy extension of Brian Day's convolution monoidal structure on a presheaf category. An easy way of seeing the truth of Hovey's observation is to recall that a tensored V-category structure on a (mere) category A corresponds to an _action_ of V on A having an appropriate right adjoint; see [Janelidze and Kelly, TAC 9 (2001), 61 - 91], foot of page 66; but the idea itself is quite old. Now V --> [D,D] easily gives [A,V] --> [[A,D],[A,D]]. Of course there is checking to do. Max Kelly. 15-Jan-2002 08:46:06 -0400,1185;000000000000-00000000