John Baez asked, on January 24, about anafunctors. As far as I know, the notion was first explicitly introduced in my paper "Avoiding the axiom of choice in general category theory", in JPAA 108 (1996), 109-173. The term was suggested by Dusko Pavlovic. Precursors occur in the work of Max Kelly, and Andre Joyal, as I explain in the paper. The concept John gives is equivalent to "saturated anafunctor" in the paper; plain anafunctor is something that generalizes "functor". The wording of the definition of "saturated anafunctor" is different from John's definition, but the equivalence is fairly straightforward. I should mention that John`s definition is a very useful formulation, especially when one wants to generalize things to higher dimensional categories, as I came to realize some time after I started studying the John Baez/James Dolan announcement on weak n-categories. In addition to the paper mentioned above, there is reference to anafunctors in "First Order Logic with Dependent Sorts", a monograph that will appear in Springer's Lecture Notes in Logic as soon as I manage to complete the necessary revisions; it is available electronically from the TRIPLES and HYPATIA (?) sites.