If is the canonical inclusion of (a skeleton of) into a Lawvere theory , then a factorization system over is a decomposition of into a distributive law of two other Lawvere theories.
Relation to distributive laws
Suppose given a category . Then to give a category equipped with an identity-on-objects functor and a factorization system over is the same as to give a distributive law between two monads on in the bicategory Prof. The two monads are the categories and , and their composite is .