- Under category

Let $C$ be a locally presentable category, and let $c$ be an object of $C$. Lets denote by $C^{/c}$ the under category, objects are maps $c\rightarrow x$ and morphisms are the evident ones. I was wondering if the category $C^{/c}$ is also locally presentable ?

- Monad

Let $T$ be a monad on a locally presentable category $C$. Under which condition the category of $T$-algebras (the category of algebras over the monad $T$) is a locally presentable? Examples are very welcome.