  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 $\mbox{Mod-} R$ 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 $\mbox{Mod-} R$
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.
