Distributive law between monads

In category theory, an abstract branch of mathematics, distributive laws between monads are a way to express abstractly that two algebraic structures distribute one over the other.

Suppose that $$(S, \mu^S, \eta^S)$$ and $$(T, \mu^T, \eta^T)$$ are two monads on a category C. In general, there is no natural monad structure on the composite functor ST. However, there is a natural monad structure on the functor ST if there is a distributive law of the monad S over the monad T.

Formally, a distributive law of the monad S over the monad T is a natural transformation
 * $$l:TS\to ST$$

such that the diagrams
 * Distributive law monads mult1.svg         Distributive law monads unit1.svg
 * Distributive law monads mult2.svg         Distributive law monads unit2.svg

commute.

This law induces a composite monad ST with
 * as multiplication: $$STST\xrightarrow{SlT}SSTT\xrightarrow{\mu^S\mu^T}ST$$,
 * as unit: $$1\xrightarrow{\eta^S\eta^T}ST$$.