Some weeks ago, Max Kelly brought to my attention his paper "Complete functors in homology I. Chain maps and endomorphisms", Proc. Cambridge Phil. Soc. 60 (1964), pp. 721-735. In section 2, "Generalities on functors", in the first two (full) paragraphs on page 723, he gives a definition of what I call "anafunctor" in my paper "Avoiding the axiom of choice in general category theory" (relatively) recently announced on the Net. He gives both definitions, the elementary one, and the one in the style of spans that I give in my paper. He introduces the concept (that he does not name) as the general form of the data which one is having at one's disposal in many situations when one wants to introduce a functor but the converting of the data into a functor requires a use of the axiom of choice. He does not attempt to develop a theory of such data; however, he does say: "one who will not admit such choices [requiring the axiom of choice] may work with the pair of honest functors S , T [in the span-style definition of "anafunctor"] in place of the dishonest functor ...". Thus, my paper is a working-out of a thirty-year old idea of Max Kelly's. In the paper, I endeavor to show that one can indeed "work with the pair S , T " without giving up the basic structure of category theory. Michael Makkai