Dear Toby, Thank you for your explanations. I really admire your elegant idea to use zig zags instead of 2-pullbacks. But I am an old fashioned mathematician and I like precise definitions with which I can (try to) prove precise results. You suggest to use zig zags. Could you please tell me what are the maps in the zig zags: anafunctors? functors? (in both cases what properties do you assume about them?), equivalences of categories? (in that case in what sense?) How do you compose your zig zags? You say "directly", do you mean by mere concatenation? Obviously there would be a huge amount of such zig zags, thus you would probably want to work up to some identification. Could you please tell me, with precision, when two such zig zags between two categories A and B should be identified? In the case of spans, using 2-pullbacks, what are the maps in your spans, when should two such spans between A and B be identified? You say, I quote you: Indeed, so one must also define natural isomorphism of equivalences If you have any difficulty, the answer is in Makkai's anafunctor paper Thank you for the tactfulness with which you point out my alleged ignorance, but I happen to know what anafunctors are. I have even given a very simple definition of them. It is still in the nLab, under my name (I have checked that recently) But due to my limited imagination, I don't know how to use anafunctors here. Could you please tell me how they can be used to decide when two spans or two zig zags are equivalent. Finally, I cannot help thinking about the poor people who are not familiar with anafunctors. Are they doomed forever to never understand what it means for two categories to be equivalent? I look forward to your answers. Very respectfully yours, Jean [For admin and other information see: http://www.mta.ca/~cat-dist/ ]