Dear Marta, The discussion of the equivalence in the nLab article you mention was added 5 days ago by me, by extracting and condensing a bit from Jean's, my, and David's emails a week ago. I thought this discussion interesting enough that it ought to be preserved. On Sat, Jan 29, 2011 at 9:45 AM, Marta Bunge <marta.bunge@mcgill.ca> wrote:

In that article, it is furthermore pointed out that each version has its advantages over the other, and that therefore both are of interest for category theory in a topos S without AC, where they generalize ordinary functors. But, even in the presence of AC, distributors (profunctors) generalize ordinary functors, a fact that I have known for 45 years, whereas anafunctors do not.

This confused me for a minute until I realized you probably meant "generalize" to mean "strictly generalize." (It is of course still true under AC that every functor is an anafunctor; the difference being rather that under AC every anafunctor is isomorphic to a functor.) As David pointed out, since the nLab is a wiki, anyone who feels the discussion currently existing there has defects should feel free to take it upon themselves to remedy those defects. Best, Mike [For admin and other information see: http://www.mta.ca/~cat-dist/ ]