Given an abelian category $\mathcal{A}$ the category of chain complexes over $\mathcal{A}$ is again an abelian category. If $\mathcal{A}$ is a Grothendieck category then the category of chain complexes over $\mathcal{A}$ is a Grothendieck category? In praticular, for a ring $R$ with unitary and the category of its left unitary modules, does this hold?

Feel free to retag.