Flat model structures and Gorenstein objects in functor categories

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