The extra structure on the endofunctor F given by hom(X x Y,Z) ---> hom(X x FY,FZ) is called a tensorial strength, for reasons deriving from enriched category theory, cf Kock: Strong Functors and Monoidal Monads, Archiv der Math. (Basel) 23 (1972), 113-120. In loc.cit., it is given in terms of X x FY ---> F(X x Y), but such data is equivalent to data of the above kind, by a Yoneda-like argument (put Z = X x Y, and apply to the identity map of X x Y). A kind of structure that implies such monoidal strength is the data of F being an indexed endofunctor (assuming the category in question has pull-backs, thus being indexed over itself). This seems to be the case for any endofunctor F on a topos, provided F is described by logical means. Anders Kock +++++++++++++++++++++++++++++++