Derived categories for Grothendieck categories of enriched functors

The derived category D [ C , V ] \bf D[\mathcal {C},\mathcal {V}] of the Grothendieck category of enriched functors [ C , V ] [\mathcal {C},\mathcal {V}] , where V \mathcal {V} is a closed symmetric monoidal Grothendieck category and C \mathcal {C} is a small V \mathcal {V} -category, is studied. We prove that if the derived category D ( V ) \bf D(\mathcal {V}) of V \mathcal {V} is a compactly generated triangulated category with certain reasonable assumptions on compact generators or K \mathbf K -injective resolutions, then the derived category D [ C , V ] \bf D[\mathcal {C},\mathcal {V}] is also compactly generated triangulated. Moreover, an explicit description of these generators is given.