I have recently asked a question about the composition of two monads, namely $\mathcal{M}_P = \mathcal{M}_C \cdot \mathcal{M}_C$. I am conjecturing that the cateogory of $\mathbb{C}$-Modules and the category of $\mathbb{N}$-Modules also live in a category of categories, namely a category of categories of modules, $\mathcal{C}_{Mod}$. Since the monads compose, I am guessing that the modules themselves compose in the category $\mathcal{C}_{Mod}$. We should, then, expect some kind of composition in this category. A first guess is that of a tensor product, so my first question is just whether this is true: monad composition induces tensor product. What is the standard name for the inducement where a composition of monads induces a tensor product of modules?

Edit: I think this is a monoidal functor between the category of modules and the category Mon_set, of monads on SET.