 Precomposition monad on a functor category

Precomposition defines a monad $T^{\ast }:[\mathcal{C},\mathcal{D}]\to [\mathcal{C},\mathcal{D}]$ by $G\mapsto GT$, with unit $G\eta$ and multiplication $G\mu$. Its Eilenberg-Moore category is canonically equivalent to $[{\mathcal{C}}_T,\mathcal{D}]$.