Flat model structures and Gorenstein objects in functor categories

We construct a flat model structure on the category Q,\,RMod of additive functors from a small preadditive category Q satisfying certain conditions to the module category RMod 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 Q,\,RMod 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.