Thanks for the comment. Since I probably wrote this back when let me say that I don’t have the leisure right now to look into this, sorry. But if you think there is a useful stronger statement, why not write out the details here.

]]>"The total chain complex is, under the Dold-Kan correspondence, equivalent to the diagonal of a bisimplicial set – see Eilenberg-Zilber theorem. As discussed at bisimplicial set, this is weakly homotopy equivalent to the total simplicial set of a bisimplicial set."

Would it be that the total chain complex is **exactly** the total(i.e. codiagonal or $\bar{W}$) simplicial abelian group of the bisimplicial abelian group under Dold-Kan correspondence? Sorry if I am wrong. ]]>