A pseudo-distributive law is a vertical categorification of a distributive law, relating two pseudomonads on a bicategory. There are various different levels of weakness that such a thing can exist at. As with ordinary distributive laws, a pseudo-distributive law governs the lifting of one pseudomonad to the Eilenberg-Moore and Kleisli bicategories of the other.
See any of the references, particularly (CHP)
If $T$ is a pseudo-commutative? 2-monad on Cat, then there is a pseudo-distributive law between $T$ and the 2-monad whose algebras are symmetric monoidal categories; see (Kelly).
Grothendieck fibrations and opfibrations on a category $C$ (or more generally an object of a suitable 2-category) are the algebras for a pair of pseudomonads. If $C$ has pullbacks, there is a pseudo-distributive law between these pseudomonads, whose joint algebras are the bifibrations satisfying the Beck-Chevalley condition; see (von Glehn).
generalized polycategories are naturally defined relative to a pseudo-distributive law on a Prof-like bicategory; see (Garner) for the canonical example of ordinary (symmetric) polycategories.
Pseudo-distributive laws involving lax-idempotent 2-monads have an especially nice form; see (Marmolejo) and (Walker).
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