To have the composition of two monads be a monad itself, we need a
distributive law natural transformation satisfying certain coherence
laws.


I'm interested in the strict 2-monad case, i.e. a strict 2-functor
equipped with unit and counit natural transformations that satisfy the
zig-zag equations on the nose.


I presume in such a case it's still possible for the distributive law
to satisfy its coherence laws only up to a modification.
If so, what coherence laws do those modifications have to satisfy?



Coherence laws when composing 2-monads
These are known as [pseudo-distributive laws](https://ncatlab.org/nlab/show/pseudo-distributive+law) and are the most common notion of distributive law of 2-dimensional monads, even when both 2-dimensional monads in question are strict (i.e. 2-monads rather than pseudomonads). These have been well studied, and there are many equivalent formulations of the coherence conditions. The most comprehensive account is Walker's [Distributive laws, pseudodistributive laws and decagons](https://arxiv.org/abs/2102.12468).


