# References for promorphisms of profunctors?

We can interpret a profunctor $F:C^{\mbox{op}}\times D \to \mbox{Set}$ between small categories as adjoining some morphisms to the category $C \cup D$ to get a new category $\tilde{F}$. Then a natural transformation between profunctors $F, G$ can be seen as a functor between $\tilde{F}, \tilde{G}$. We can also look at profunctors between $\tilde{F}, \tilde{G}$, but I don't know what such things are called--searching Google for "pronatural transformation" gives no hits, and "natural pro-transformation" gives a single hit, S. Yokura's paper. Nothing for "cocontinuous natural transformation" or "natural cocontinuous transformation" either.

Are these studied under some other name?

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I found renyi.hu/~aladar/MrtCtx.pdf which unfortunately seems to use funny terminology. He calls the collage of a profunctor a directed bridge, and talks about morphisms of bridges, which correspond to natural transformations. Also www.dima.unige.it/~grandis/Dbl.Cahiers.pdf may be helpful. –  David Roberts Jan 1 '11 at 4:45