Hi, I need to use the following result about the equivalence between the existence of a fibration and the existence of a right adjoint to an opportune functor between comma categories: ----------------- Let A and B be categories and let U:A-->B be a functor. Let C denote the comma category A arrow (whose objects are the arrows of A) and let D denote the comma category B over U (whose objects are arrows in B from a generic object b into U(a), together with a). Then U induces a functor V:C-->D, that basically is U (as both objects and arrows in C are arrows in A too) and is defined as follows: * V(f:a-->a')=(U(f):U(a)-->U(a'),a') for each object f:a-->a' in A; * V(g,g')=(U(g),g') for each arrow (g,g'):(f1:a1-->a'1)----->(f2:a2-->a'2) in A. Then U is a fibration iff V has a right adjoint G:D-->C s.t. G;V is the identity on D and the counit of the adjunction is the identity natural transformation. ----------------- Now, I know (or at least I believe) that the result holds, as I have a proof for it; but a collegue said to me that this is a known result and suggested to look at the works by Street in the Sydney Category Seminar. Unfortunately, being quite ignorant about 2categories, I find very hard even to understand in which sense the result I want is there. Does anybody know an easier reference, possibly not involving 2categories? Thanks Maura Maura Cerioli DISI - Dipartimento di Informatica e Scienze dell'Informazione Viale Benedetto XV, 3 16132 Genova ITALY