The Comprehensive Factorization of a Functor

In this article we show that every functor has a factorization into an initial functor followed by a discrete O-fibration and that this factorization is functorial. Size considerations will be ignored but may be easily filled in; we assume the existence of a category of sets large enough to dwarf any given finite number of categories. There is an analogy between the category Set of sets and the category Cat of categories which is partly explained by the observation that each is a category of types for a suitable hyperdoctrine. A hyperdoctrine (Lawvere [4]) consists of a category Tof types and a functor P: T -• Cat satisfying conditions. The comprehension schema (also see [4]) is expressed by a pair of adjoint functors