<< Let C be a category with finite products, and let F : C -> C be a functor, along with a natural transformation F*(X,Y,Z) : hom(X x Y, Z) --> hom(X x FY, FZ) that preserves composition and identities in the appropriate manner. (The idea is that this gives a behaviour for F on collections of arrows from Y to Z indexed by X.) >> It seems that Ralph Loader has rediscovered an important case of what are called Enriched Functors, see eg Max Kelly's book. He also notes that these are distinct from Internal Functors. Internal and Enriched are both included under and interact within the "indexed" (I would prefer "parameterized") theory treated in another important book: Springer LNM 661 ++++++++++++++++++++++++++++++