Diagrams of categories, which are functor categories from a fixed index category to 2-categories of categories equipped with extra structure, are widely applied in sheaf theory of manifolds and algebraic varieties, topos theory, and representation theory of categories. In this talk I will describe some preliminary knowledge on diagrams of abelian categories and representations (modules) over diagrams, and introduce several functorial constructions on these module categories over diagrams. These results will lay the foundation for our further investigation on cotorsion theory in functor categories. This is a joint work with Liping Li, Li Liang, and Nina Yu.