Dear Categorists, I'm interested in looking at the following type of thing: Start with an essentially small category, T, and look at the category whose objects are triples (P,F,c) where: P is a contravariant functor T -> Set, F is a covariant functor T -> Set and c is a natural transformation from P x F to the Hom bi-functor. Morphisms are pairs of natural transformations P_1 -> P_2 and F_2 -> F_1 that intertwine the natural transformations c_1 and c_2. One could also enrich the whole structure. Has this cropped up anywhere before? If so, what is it called and where can I learn about it? If not, what shall I call it? If this is something standard then please pardon my ignorance. I'm fairly new to _real_ category theory and am still just learning the basics. Thanks, Andrew Stacey