  We prove a general criterion which guarantees that an admissible subcategory
$\mathcal{K}$ 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 $\mathcal{K}$ has a strongly unique dg enhancement,
applying the recent results of Canonaco, Neeman and Stellari. We apply this
criterion to the Kuznetsov component $\mathop{\mathcal{K}u}(X)$ when $X$ is a
cubic fourfold, a Gushel--Mukai 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 $\mathop{\mathcal{K}u}(X)
\xrightarrow{\sim} \mathop{\mathcal{K}u}(X')$ are of Fourier--Mukai type when
$X$, $X'$ belong to these classes of varieties, as predicted by a conjecture of
Kuznetsov.
