I recently asked a question and it contains a reference to the monads $\mathcal{M}_N$ and $ \mathcal{M}_C$, which are the multiset and $\mathcal{C}$-module monad, respectively. I want to define a distributive law for the composition $\mathcal{M}_C \cdot \mathcal{M}_N$. I am guessing there is a large freedom to define the distributive law and I won't know how to choose one. Basically, I want a monad that is a complex combination (c-module) of numbers of things of different types (multiset). I am asking for any help, hints, or complete solutions.
A distributive law is necessary to be able to compose the functors of two monads and generate a new monad from the composition. That is what I am looking for, the monad derived fromt the composition. At the same time, it would be nice to know the modules that result from the composition of $ \mathbb{N}$-modules and $ \mathbb{C}$-modules. I think that composition is a tensor product. These would be complex combinations of multisets.