Adjointness in descent theory (Q678831)

Every functor is isomorphic to a composite of a functor which is essentially surjective on objects and a functor which is full and faithful. The abstraction of this to a factorization system in a bicategory was a basic ingredient of the reviewer's paper [``Characterization of bicategories of stacks'', Lect. Notes Math. 962, 282-291 (1982; Zbl 0495.18007)], where the notion of kernel was made unduly ornate. A simpler kernel related to lax descent is sufficient. The present paper can be seen as a development of these ideas with emphasis on the biadjunction between kernel and quotient operations.