Eilenberg–Moore category

Summary

The Eilenberg–Moore category of a monad $T$ on a category $\mathcal{C}$ is the category of algebras over $T$ and their morphisms. It comes with a forgetful functor to $\mathcal{C}$ and a free algebra functor, and these form an adjunction whose monad is $T$ again. It is the terminal adjunction that gives $T$, and so it shows that every monad comes from an adjunction.