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 \({{\,\mathrm{Mod-}\,}}R\) 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 \({{\,\mathrm{Mod-}\,}}R\) and reprove a result ([4, Theorem 3.18]) of Estrada and Virili.
