The monoid description of a monad requires an endofunctor category

When the endofunctors of $\mathcal{C}$ and their natural transformations form the functor category $[\mathcal{C},\mathcal{C}]$ (Functor category $[\mathcal{C},\mathcal{D}]$), composition makes that category monoidal, and the data and equations of Monad on a category say exactly that $(T,\eta ,\mu )$ is a monoid object in it. In particular this description is available for a small $\mathcal{C}$; If $\mathcal{C}$ is small and $\mathcal{D}$ is locally small then $[\mathcal{C},\mathcal{D}]$ is locally small; if both are small it is small supplies the corresponding smallness and local-smallness conclusions.

For an arbitrary large $\mathcal{C}$, the library's convention treats endofunctors and natural transformations only as metatheoretic shorthand and does not form them into a category. The monad definition itself remains meaningful there, but the phrase “monoid object in the endofunctor category” is used only when that category exists.