Suppose $\mathcal{C}$ is a dg category (over some base) with all colimits. We say that $X\in \mathcal{C}$ is a generator if $\mathcal{C}$ is equivalent to $\operatorname{End}_\mathcal{C}X$-modules (via the Yoneda functor). My question: are there some "niceness" conditions we can impose on the pair $(\mathcal{C}, X)$ which guarantee that all deformations of $X$ (i.e. all fibers of a suitably perfect sheaf of objects of $\mathcal{C}$ over a curve with special fiber $X$, say) are still generators?