Two notions of modules over a monad

Question

If a module over a monad $T: C \rightarrow C$ is an object $c \in C$ together with a map $Tc \rightarrow c$ satisfying associativity conditions

(as in https://ncatlab.org/nlab/show/algebra+over+a+monad, this n-lab page doesn't use the word module though but use algebra instead, I saw module being used in this n-lab page.)

then what do we call a module over $T$ in the sense of monoidal categories? I.e If we define $F \otimes G = F \circ G$ to be the tensor operation on $\text{End}(C)$ making it into a monoidal category then monoids in $\text{End}(C)$ are monads and a module (in the sense of monoidal categories) over a monad $T$ is an endofunctor $A: C \rightarrow C$ together with a natural transformation $TA \rightarrow A$.

Is this why we don't often call modules over $T$ modules but algebras instead to distinguish them from modules in the monoidal category sense? I'm confused.

Thanks for answering in advance!

Answer 1

The endofunctor sense is simply not all that common. But there's no reason not to use "module" for both. In fact, they are both special cases of the following notion of module: $\mathrm{End}(C)$ acts on the functor category $[D,C]$, and any time a monoidal category $M$ acts on a category $K$, the monoids of $M$ may have modules from $K$.

Thus we can let the monad $T$ act on a functor $F:D\to C$, which gives the usual notion of module (or algebra) over a monad when $D=*$, while it gives your notion when $D=C$. The use of general $D$ is important for considering monads in a more general 2-category than $\mathrm{Cat}$, where the terminal object may not have the strong generation properties it does in $\mathrm{Cat}$.