TITLE: About the category of chain complexes and Grothendieck categories.
QUESTION [5 upvotes]: 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.

REPLY [6 votes]: Yes, this is stated e.g. on page 3 of Hovey: Model category structures on chain complexes of sheaves.