In the context of Grothendieck toposes, the Comparison Lemma says that if $(\mathbb{C}, \mathcal{J})$ is a small site and $\mathbb{B} \hookrightarrow \mathbb{C}$ is a full subcategory with induced Grothendieck topology $\mathcal{K}$ such that every object of $\mathbb{C}$ has a covering by objects of $\mathbb{B}$, then we have an equivalence of categories $\mathsf{Sh}(\mathbb{C}, \mathcal{J}) \simeq \mathsf{Sh}(\mathbb{B}, \mathcal{K})$.
Does a similar result hold (more generally) for categories of separated presheaves, i.e. Grothendieck quasitoposes of the form $\mathsf{Sep}(\mathbb{C}, \mathcal{J})$, rather than categories of sheaves?