Pseudoskew category algebras and modules over representations of small categories

Let $\mathcal{C}$ be a small category and let $R$ be a representation of the category $\mathcal{C}$, that is, a pseudofunctor from a small category to the category of small preadditive categories. In this paper, we mainly study the category $Mod-R$ of right modules over $R$. We characterize it both as a category of the Abelian group valued functors on $Gr(R)$ and as a category of modules over a new family of algebras: the pseudoskew category algebras $R[\mathcal{C}]$, where $Gr(R)$ is the linear Grothendieck construction of $R$. Moreover, we also classify the hereditary torsion pairs in $Mod-R$ and reprove 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.