This refers to the original monad-monad distributive law (Beck, 1969); variants are available.
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
- almost distributive law
References