Does the community have any reaction to the following observation, that we have made these last few days? Let A and B be 2-categories, and F,G : A ---> B be 2-functors. There is a notion of 2-natural transformation t : F ---> G; such a t is not only to be natural, in the usual sense, with respect to 1-cells, but to have the appropriate property with respect to 2-cells as well. Certainly mere natural- ity does not imply 2-naturality, as one sees on taking for A the free-living 2-cell. Yet it DOES imply 2-naturality if A admits tensor products with the arrow-category 2 (or dually cotensor products with 2). A consequence is that the forgetful functor, from the category of finitary 2-monads on a locally-finitely-presentable 2-category A to the category of finitary monads on the underling category of A, is a fully-faithful functor with both adjoints. In particular, a finitary monad on Cat or such-like has at most one enrichment (although usually none) to a 2-monad; and similarly for the more common case of enrichability over the closed category of groupoids. John Power and Max Kelly, at Sydney - 24 Jan. ======================================