@TimCampion Thanks, sorry can you specify the result you're using for "compute the left adjoint on representables pretty easily to see it agrees with the dold kan functor"? I tried to use the fact that every colimit-preserving functor out of a presheaf category has a right adjoint (determined by value on representables), but DK is the left adjoint. Or are you using that Ch(CMon) is embedded in a presheaf category...?

@ChrisSchommer-Pries I agree that your construction gives a left inverse to $\mathrm{DK}$. To prove $\mathrm{DK}$ is full, I seem to need that your construction is a faithful functor. Is this obvious/true?

Ah, of course, thank you both for the clarification! In that case I believe the original logic stands, and Chris's suggestion answers my question allowing me to generalize full faithfulness to any semiadditive category.

@MahdiMajidi-Zolbanin Sorry to revive this old question, I don't understand the deduction from Corollary 4.1.1, which implies that the restriction of $M_{(X,Y)}$ to the punctured spectrum of $\mathbb{C}[X,Y]_{(X,Y)}$ is a trivial bundle. How do we know $M_{(X,Y)}$ is trivial on the entire spectrum?

@ChrisSchommer-Pries Update: I believe you are correct. Some confusion stemmed from different notation between Lurie and Goerss-Jardine. Regardless, your functor is a proper left inverse to what I've called $\mathrm{DK}$ and I will be happy if you post as an answer.

That is indeed a chain complex and functorial. It occurs to me that the original Dold-Kan defines this complex with differential $(-1)^{n}d_n$ only so that the natural map into the Moore complex is a chain map, which is needed for the other direction of equivalence. In any case, I think your functor $\widetilde{N}$ is indeed a left inverse to $\mathrm{DK}$ via the usual proof. In this case I hope you will post this as an answer.

Thanks, for the nice reference. Unfortunately the main object of study in that paper is a functor back sA -> Ch(A), which a priori we don’t have here.