A definition of the usual monad-monad distributive law is in string diagrams for distributive laws.
Proposition. Assume we find two monads whose composition is a monad. If the multiplication and unit of the composite monad come from a candidate distributive law, then that candidate is indeed a distributive law.
Examples and counterexamples.
- up monad and down comonad of an adjunction collage
- normalization is almost a distributive law
- weak distributive law
References