Dear Peter, In one of my mails I mentioned the following result, which I thought to be original: Proposition: Let P: X --> S be a fibration. The functor P is final iff all its fibers are connected From your answer to that mail, dated December 29, I quote: "please don't deceive yourself that this is a new result. It is a (very) special case of the theorem of Street and Walters ("The comprehensive factorization of a functor", Bull. Amer. Math. Soc. 79, 1973) that the pair (final functors, discrete fibrations) forms a factorization structure on Cat. It's true that this result is not stated in the Elephant (why on earth should it be?), but the Street--Walters factorization (for internal categories) is treated in section B2.5." I tried to prove that my proposition was a consequence of the theorem of Street-Walters which you quoted in you mail, but did not succeed. Then I consulted their original paper, hoping to find there more details which would help me to find a proof. Again in vain. I'm quite sure that you're right, and that my inability to get a proof is entirely due to my mathematical limitations. Thus I'd really be very grateful, if you'd give me a proof, or even a sketch of a proof, that my proposition is an easy consequence of the theorem of Street and Walters. Many thanks in advance and best regards, Jean [For admin and other information see: http://www.mta.ca/~cat-dist/ ]