A characterization of modules over dg-representations of small categories

Let $\mathcal{C}$ be a small category and let $R$ be a dg-representation of the category $\mathcal{C}$, that is, a pseudofunctor from a small category to the category of small dg $k$-categories, where $k$ is a commutative unital ring. In this paper, we mainly study the category $\mbox{Mod-} R$ of right modules over $R$. We characterize it as an ordinary category of dg-modules over a (differential graded) dg-category $Gr(R)$, where $Gr(R)$ is the linear Grothendieck construction of $R$. This characterization generalizes the Theorem 3.18 of the paper (S. Estrada and S. Virili. Cartesian modules over representations of small categories. Adv. in Math. 310: 557-609, 2017) of Estrada and Virili to the dg-category context. Furthermore, as some applications of the main characterization theorem, we classify the hereditary torsion pairs, (split) TTF triples and Abelian recollements in $\mbox{Mod-} R$ respectively.