Consider the Moore complex functor $M : \mathsf{sAb} \to \mathrm{Ch}^+(\mathsf{Ab})$, where the complex associated to a simplicial abelian group $A$ is $MA_n = A_n$ with boundary $\partial = \sum_{j=0}^n (-1)^j d_j$. If we look at the normalized subcomplex $NA_n = \ker d_0 \cap \ldots \cap \ker d_{n-1}$ then we get another functor $N : \mathsf{sAb} \to \mathrm{Ch}^+(\mathsf{Ab})$, which the Dold-Kan Correspondence says is an equivalence of categories (and further a Quillen equivalence). The fact that we need to normalize here makes me think $M$ itself is not an equivalence $\mathsf{sAb} \to \mathrm{Ch}^+(\mathsf{Ab})$, but I don't see how to prove it. I'm also wondering whether $M$ even has an adjoint (on either side). It seems like $M$ preserves limits, since limits in both $\mathsf{sAb}$ and $\mathrm{Ch}^+(\mathsf{Ab})$ are computed degreewise, and $(MA)_n = A_n$.
Edit: I checked that $M$ preserves all limits and colimits, and according to the nlab $\mathsf{sAb}$ is a "total" and "cototal" category (as it's the category of abelian sheaves on a site) so an adjoint functor theorem ensures $M$ has adjoints on both sides. What are these adjoints? Do they have a nice interpretation?