I would like to know if there is a partial monoidal product on the category of monads on Set. I want this partial monoidal product to "handle" monad composition which we understand exists when there is a distributive law. I want to be able to do familiar monoidal moves, which I will explain. Given these monads, $M_S = \langle S , \eta_S, \mu_S \rangle$ and $M_T = \langle T , \eta_T, \mu_T \rangle$, suppose you have a distributive law, $\lambda: ST \rightarrow TS$. The underlying endofunctor of the composite is $ST$. Now say you have some monad maps $F_L: M_S \rightarrow Q$ and $F_R: M_T \rightarrow Q'$. I want a monoidal product, where I can write $M_S \otimes M_T$, and so that I can do the following:

$$F_L \otimes I : M_S \otimes M_T \rightarrow Q \otimes M_T$$ $$I \otimes F_R : M_S \otimes M_T \rightarrow M_S \otimes Q'$$

I guess the problem is that there can be more that one distributive law given two monads. The fact that there can be zero distributive law is handled by this product being partial. But the other way around means that monad composition cannot be subsumed in a product. Has anyone seen any monoidal products defined on the category of monads on Set?

There is some language I am seeing but I don't understand: "composite along a distributive law" and "connection between the distributive law and $\otimes$". People seem to be implying that if I give a"connection" between the distributive law and the tensor, this might be an intelligible question. How do I give a connection between the distributive law tensor product?