I don't have a grasp of model categories. I asked the question through Quillen equivalences in order to make it as general as possible. This might be too general to answer and/or might be above my reach. I just have a grasp of Dold-Kan correspondence in the classical setting and wanted to have an idea about this question.
What I really want to ask is the following:
By Dold-Kan we have $(\Gamma \dashv N) : sAb \stackrel{\overset{\Gamma}{\leftarrow}}{\underset{N}{\to}} Ch_\bullet^+$ an equivalence of categories where for $A \in sAb$, ${\pi}_n(A,0) \simeq H_n(NA, \mathbb{Z})$.
Let $(F \dashv G) : sAb \stackrel{\overset{F}{\leftarrow}}{\underset{G}{\to}} Ch_\bullet^+ $ be an equivalence of categories such that ${\pi}_n(A,0) \simeq H_n(GA, \mathbb{Z})$.
I want to investigate if $F$ and $G$ are in some way equivalent to $\Gamma$ and $N$. One can also exclude the condition of $ {\pi}_n(A,0) \simeq H_n(GA, \mathbb{Z}) $ and look at this new question too, but this looks like wildly different than the other two questions, so this is just a side-question. It would be nice to have a reference to see how the "shifting the indices one down" operation in chain complexes corresponds to a functor for simplicial abelian groups, but as I said before, I don't want to make this a mega question. However, in usual references for Dold-Kan, I couldn't find something related to other possible equivalences in general.
Any reference, hint, idea regarding these questions would be nice. Thanks!
Since equivalences of categories are cocontinuous, it suffices to show that F and G restrict to an equivalence of categories on chain complexes of the form Z[n] and simplicial abelian groups of the form Γ(Z[n]), and these restrictions are isomorphic to Γ and N respectively.
Indeed, Z[n] and Γ(Z[n]) are uniquely characterized by the property that their nth homology respectively homotopy group is Z and the other groups vanish. This proves the first claim.
For the second claim, observe that morphisms from Z[m] to Z[n] vanish if m≠n and are canonically isomorphic to Z if m=n. Likewise for Γ(Z[m]) and Γ(Z[n]). This means that any family of isomorphisms F(Z[n])→Γ(Z[n]) is automatically natural and likewise for G and N.
Hence, F is isomorphic to Γ and G is isomorphic to N.