 Pointwise monad on a functor category

For a monad $T$ on $\mathcal{C}$ and a category $\mathcal{D}$, postcomposition defines a monad $T_{\ast }:[\mathcal{D},\mathcal{C}]\to [\mathcal{D},\mathcal{C}]$ whose unit and multiplication are pointwise. Its Eilenberg-Moore category is canonically equivalent to $[\mathcal{D},{\mathcal{C}}^T]$.