Presenting Distributive Laws

Distributive laws of a monad \(\mathcal{T}\) over a functor F are categorical tools for specifying algebra-coalgebra interaction. They proved to be important for solving systems of corecursive equations, for the specification of well-behaved structural operational semantics and, more recently, also for enhancements of the bisimulation proof method. If \(\mathcal{T}\) is a free monad, then such distributive laws correspond to simple natural transformations. However, when \(\mathcal{T}\) is not free it can be rather difficult to prove the defining axioms of a distributive law. In this paper we describe how to obtain a distributive law for a monad with an equational presentation from a distributive law for the underlying free monad. We apply this result to show the equivalence between two different representations of context-free languages.