Dear Peter,
Incidentally, in reply to a comment in Marta Bunge's posting, I did distribute draft copies of the Elephant to a number of colleagues, and invite their comments, before it was published. None of them picked up this particular point -- though I am not blaming them for that; the fault was of course mine.
I have now read in the Preface to the Elephant that you did indeed circulate a draft to "a number of colleagues (including Ieke Moerdijk and Andrew Pitts, as well as Martin Hyland, Anders Kock and Gavin Wraith) for their comments and suggestions". Apologies for my swift comment. I had gotten a different impression from the reaction of several participants to the Fields Institute Workshop on Galois Theory etc, where the first two volumes of the Elephant were exhibited, including comments by Bill Lawvere and several other people whose work was prominently represented in the book. Aurelio Carboni and I perused quickly section B4.5 on the symmetric monad, and were satisfied on the spot with your account of it, but it did not occur to me to look up section B1.5, even though my paper with Bob Pare on stacks was cited at the end of the section. Had I done so, I would have pointed out that Proposition 1.5.5 is due to Benabou and Roubaud as we ourselves had pointed out in our paper. I take this opportunity to apologize to Jean Benabou for stating his theorem with Jacques Roubaud in the context of indexed categories and not on that of (bi)fibrations without warning the reader, but I also want to point out that, since the proof itself only refers to two fibers and a transition map between them, it is equally meaningful in both contexts. I believe that I only became aware of the gross difference afterwards, when lecturing on stack completions at the Benabou Seminar in Paris (whenever that was -- 1979?). I still think now that it is easier to work with the indexed presentations of fibrations than with the fibrations themselves, without attempting to turn this into a philosophical statement of any kind. Beck's Tripleability theorem is indeed useful in the applications of the Benabou-Roubaud theorem, and among them are those we give in the Cahiers paper on stacks, and in its sequel by myself on stack completions. Perhaps the relabelling of the Chevalley condition as "the Beck condition" led to some confusion as to whether the Benabou-Roubaud theorem had been proved (also) by Jon Beck? I myself never heard him speaking on this, and never saw any draft written by him of a proof of this theorem. Sadly, we cannot consult Jon himself on this issue, so we might as well drop it. Perhaps you would consider, prior to publication of Volume III of the Elephant, enlarging your list of commentators to include at least those whose work you include in some form or other. With best regards, Marta