It seems to be a well-known fact that homotopy (co)limits of nonnegatively graded (co)chain complexes in (Grothendieck) abelian categories can be computed by using the Dold-Kan correspondence to pass to double complexes and then applying the totalization functor for double complexes, possibly applying the truncation functor afterward. For example, this is claimed (without proof) in Dugger's notes on homotopy colimits, see Section 16.8 in http://math.uoregon.edu/~ddugger/hocolim.pdf.

Is there a written proof of this result in the literature?

What about the case of unbounded chain complexes?