Cartesian morphism ~~> fibration

Dear all, I'm trying to find a reference for the following result, if indeed it is true. Let X1->B and X2->B be fibrations and F:X1->X2 a cartesian functor over B. Then F factors on the nose as X1 -> X1' -> X2 (as functors over B) such that X1->X1' is an equivalence and X1'->X2 is a fibration. I know it is true if X1 and X2 are fibred in groupoids, this construction is in the Stacks Project. But the general case? Thanks, David [For admin and other information see: http://www.mta.ca/~cat-dist/ ]