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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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    $\begingroup$ 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. $\endgroup$
    – David Roberts ♦
    Commented Jan 1, 2011 at 4:45
  • $\begingroup$ To spell this out in different terminology, what you are describing are horizontal morphisms between cotabulators in the pseudo-double category of profunctors (taking horizontal morphisms to be the profunctors). All natural transformations between cocontinuous functors are "cocontinuous" in the appropriate sense, so I don't see that this corresponds to the concept you are looking for. Did you unwind the data of such a profunctor? That might make it easier to find other references. $\endgroup$
    – varkor
    Commented Jul 6, 2022 at 12:38
  • $\begingroup$ For later readers, David's first link is to On Morita Contexts in Bicategories. $\endgroup$
    – varkor
    Commented Jul 6, 2022 at 12:39

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