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.

In the above, “homotopy (co)limit” is used in the abstract ∞-categorical sense, i.e., the homotopy terminal (respectively initial) object in the ∞-category of (co)cones. It can be presented as the appropriately derived functor of the ordinary (co)limit functor in the setting of (stable) model categories or as the quasicategorical (co)limit in the setting of stable quasicategories.

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

What about the case of unbounded chain complexes?