$t$-Structures on stable derivators and Grothendieck hearts

We prove that given any strong, stable derivator and a $t$-structure on its base triangulated category $\mathcal{D}$, the $t$-structure canonically lifts to all the (coherent) diagram categories and each incoherent diagram in the heart uniquely lifts to a coherent one. We use this to show that the $t$-structure being compactly generated implies that the coaisle is closed under directed homotopy colimit which in turns implies that the heart is an (Ab.$5$) Abelian category. If, moreover, $\mathcal{D}$ is a well generated algebraic or topological triangulated category, then the heart of any accessibly embedded (in particular, compactly generated) $t$-structure has a generator. As a consequence, it follows that the heart of any compactly generated $t$-structure of a well generated algebraic or topological triangulated category is a Grothendieck category.