Derived categories of hearts on Kuznetsov components

We prove a general criterion that guarantees that an admissible subcategory. of the derived category of an abelian category is equivalent to the bounded derived category of the heart of a bounded t- structure. As a consequence, we show that. has a strongly unique dg enhancement, applying the recent results of Canonaco, Neeman, and Stellari. We apply this criterion to the Kuznetsov component...(X) when.. is a cubic fourfold, a GM variety, or a quartic double solid. In particular, we obtain that these Kuznetsov components have strongly unique dg enhancement and that exact equivalences of the form...(X) ->(X) are of Fourier-Mukai type when..,.. ' belong to these classes of varieties, as predicted by a conjecture of Kuznetsov.