Starting with a Grothendieck category and a torsion pair in , we study the local finite presentability and local coherence of the heart of the associated Happel-Reiten-Smal{\o} -structure in the derived category . We start by showing that, in this general setting, the torsion pair is of finite type, if and only if it is quasi-cotilting, if and only if it is cosilting. We then proceed to study those for which is locally finitely presented, obtaining a complete answer under some additional assumptions on the ground category , which are general enough to include all locally coherent categories, all categories of modules and several categories of quasi-coherent sheaves over schemes. The third problem that we tackle is that of local coherence. In this direction we characterize those torsion pairs in a locally finitely presented for which is locally coherent in two cases: when the tilted t-structure in is assumed to restrict to finitely presented objects, and when is cogenerating. In the last part of the paper we concentrate on the case when is a category of modules over a small preadditive category, giving several examples and obtaining very neat (new) characterizations even in this more classical setting, also underlying connections with the notion of an elementary cogenerator.