We consider TTF-classes which are not triangulated but just cosuspended, i.e., subcategories V of T for which there is a (nondegenerate) t-structure (U,V) and a co-t-structure (V,W). Examples of such TTF-triples (U,V,W) arise from compactly generated t-structures or from certain cosilting t-structures in derived module categories. In this talk, we show that in a compactly generated triangulated category, a large class of these TTF-triples can be parametrised by pure-injective silting objects. Moreover, it turns out that the heart of such a cosilting t-structure is a Grothendieck category and, as a consequence, nondegenerate compactly generated t-structures have Grothendieck hearts. This talk is based on joint work with Lidia Angeleri Hügel and Frederik Marks.