Let $\mathcal{C}$ be an additive category and let $N_{dg}(\mathcal{C})$ be the differential graded nerve of the differential graded category $Ch(\mathcal{C})$. This is a stable $\infty$-category.

Note that $Ind(\mathcal{C})$ is naturally an additive category. Then is it true that $$Ind(N_{dg}(\mathcal{C})) \simeq N_{dg}(Ind(\mathcal{C}))?$$

In general, when does the nerve functor $N_{dg}$ commute with "categorical" constructions that can be performed in both the setting of $dg$-categories and the setting of $\infty$-categories?

boundedcomplexes of finite-dimensional vector spaces. $\endgroup$ – Pavel Safronov Oct 15 '18 at 7:31