I'm looking for a reference for the following fact:

take a simplicial chain complex $ X:\Delta^{op}\to Ch_{\geq 0}(\mathcal A)$ for $\mathcal A$ a nice abelian category (say, cocomplete with enough projectives, although I'm willing to add more hypotheses, because the $\mathcal A$ I want to use it for is the category of connective modules over some connective dga ); then a model for the homotopy colimit of $X$ is the total complex of the bicomplex associated to $X$.

I know a proof (I haven't checked the details so I'm not sure it works for an arbitrary dga - it works at least for discrete rings), so that's not what I'm looking for (except if you have an especially short and elegant one, then it wouldn't hurt to see it); I'm mostly looking for a reference.

I know the result is mentioned in Dugger's *A primer on homotopy colimits* (proposition 19.9) but there seems to be no proof in there - so I'll add the criterion that the reference should contain a proof.

This may be related to this question, which relates the total complex and the diagonal - since there is an answer with a reference there, it would also suffice to provide a reference for the fact that the diagonal is a model for the homotopy colimit (actually, this would be enough for other reasons : one can use the diagonal model for simplicial objects that land in $\mathcal A$, and then use homotopy cofinality of $\Delta^{op}\to \Delta^{op}\times \Delta^{op}$ to get the result for an arbitrary simplicial chain complex).

For the latter, I know references for simplicial sets, but not for simplicial $\mathcal A$-objects (and in the case of a discrete ring, one may use this as well via the usual adjunction).

The answers given here seem to be unsatisfactory given the comments below.

Here, the question itself provides a sketch of proof for $\mathbb Z$ which I think can be adapted to the general case, but the adjunction that is mentioned does not seem crystal clear to me (if you could explain it, that would also be great) and it's not a reference.