Dear all, Anders Kock's reply to Yemon Choi gives me a good opportunity to pose a question which I have been meaning to ask the list for a while:
The V-enrichment ("strength") of an endofunctor T on V can be encoded without reference to the closed structure of V as a transformation T(A)@B-->T(A@B) ("tensorial strength", introduced in [4]).
This notion of "tensorial strength" is just a special case of what I would call "an action of a monoidal functor on a (mere) functor". Specifically, it is a right-action of the identity monoidal functor on the functor T. In general, given a monoidal functor M:V-->W and a functor T:V-->W, a right-action of M on T should be a n.t. of the form T(A)@M(B)-->T(A@B) satisfying the obvious associativity and unitality axioms. For instance, if we regard a G-graded algebra as a monoidal functor G-->Vec, then a right-action of this on a mere functor G-->Vec is precisely the same thing as a G-graded right-module. [Here the monoid G (G can also stand for grading-object!) is considered as a discrete monoidal category.] I have always assumed that this concept is well-known, but I haven't succeeded in finding a reference in the literature for it... perhaps some of the more well-read readers of this list could help me out? Cheers, Jeff. P.S. Upon reviewing [4], I see that there is a more general notion of tensorial strength which can be applied to a functor A-->B whenever A and B are tensored over some monoidal category V. There is a similar adaptation of the notion of action of a monoidal functor V-->W to functors A-->B whenever A is tensored over (or I would say, acted on by) V, and B over (by) W.
[4] Strong functors and monoidal monads, Archiv der Math. 23 (1972), 113-120.