Hearts of set-generated t-structures have a set of generators

We show that if $\alpha$ is a regular cardinal, $\mathcal{D}$ is an $\alpha$-compactly generated triangulated category, in the sense of Neeman \cite{N}, and $\tau$ is a t-structure in $\mathcal{D}$ generated by a set of $\alpha$-compact objects, then the heart of $\tau$ is a locally $\alpha$-presentable (not necessarily Ab5) abelian category. As a consequence, in a well-generated triangulated category any t-structure generated by a set of objects has a heart with a set of generators.