Flat model structures and Gorenstein objects in functor categories

We construct a flat model structure on the category ${}_{\mathcal{Q},R}\mathsf{M}\mathsf{o}\mathsf{d}$ of additive functors from a small preadditive category $\mathcal{Q}$ satisfying certain conditions to the module category ${}_R\mathsf{M}\mathsf{o}\mathsf{d}$ over an associative ring $R$, whose homotopy category is the $\mathcal{Q}$-shaped derived category introduced by Holm and Jorgensen. Moreover, we prove that for an arbitrary associative ring $R$, an object in ${}_{\mathcal{Q},R}\mathsf{M}\mathsf{o}\mathsf{d}$ is Gorenstein projective (resp., Gorenstein injective, Gorenstein flat, projective coresolving Gorenstein flat) if and only if so is its value on each object of $\mathcal{Q}$, and hence improve a result by Dell’Ambrogio, Stevenson and Šťovíček.