I'm looking for a reference for the following lemma on composition of fibrations
Let p:E->B and r:B->A be fibrations and D be a subcategory of A-> stable under pullback. If r has indexed products along morphisms in D and if p has indexed products along all cartesian maps above something in D then the composition rp has indexed products along D, too and the Beck-Chevalley condition holds.
Martin Hofmann, Edinburgh
I suppose you want to show 1) if r and p have enough indexed products, then rp has them; 2) if r and p satisfy the Beck-Chevalley (what-in-the-world) condition, then rp does - and then all this relativized with respect to a family of arrows D. ad 1) To construct the indexed products for rp, one needs NOT only enough indexed products in r and in p, but p must ALSO satisfy a weak form of the Beck-Chevalley-what-in-the-world condition. (I suppose the term "indexed products", as used in the query, does not include this condition - since it is singled out at the end.) The construction is in my thesis: Predicates and Fibrations, Rijksuniversiteit Utrecht 1990, prop. II.3.73 I think I had an example of two poset morphisms, which were fibrations, showing that the indexed products alone do not suffice. ad 2) A direct derivation of the Beck-Chevalley for the products in rp from this condition for the products in r and in p - seemed to me rather involved, because one had to chase those vertical-cartesian spans - the arrows of the opposite fibration (which is where the indexed products come about). It was simpler to forget the indexed products, and use my characterisation of the Beck-Chevalley-world condition for fibrations in general: Categorical interpolation: descent and the Beck-Chevalley-world condition without direct images, Springer Lecture Notes in Mathematics 1488 (The characterisation in this paper is not as simple as it could be, but I think it can be used.) Regards, Dusko Pavlovic P.S. Sorry. ==============================================================================