Bi-presheaves

[Andrew Stacey <andrew.stacey@math.ntnu.no>]

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