The Dold-Kan correspondence gives an equivalence of categories between $SAb$, the category of simplicial abelian groups, and $Ch_{\geq 0}$, the category of non-negatively graded chain complexes of abelian groups.

Recall that the equivalence in the direction $C: SAb \rightarrow Ch_{\geq 0}$ is most naturally given $C(A)_{n}=A_n/(s_{0}A_{n-1}+\cdots s_{n-1}A_{n-1})$ (simplices modulo degenerate simplices) and taking the differential to be $d=\sum (-1)^{i}d_{i}$. The equivalence in the other direction $\Gamma: Ch_{\geq 0} \rightarrow SAb$ is most naturally given by $\Gamma(C)_{n}=Ch_{\geq 0}(C(\mathbf{Z}\Delta[n]),C)$ (the group of chain maps) where $\Gamma(C)$ gets a simplicial structure by pulling back the comsimplicial structure on the collection of complexes $C(\mathbf{Z}\Delta[n])$.

The reason that I say this is the most natural is because fits the standard pattern of constructing an adjunction between a presheaf category and another category. Namely, there is a functor $\Delta \rightarrow Ch_{\geq 0}$ sending $[n]$ to the complex of non-degenerate simplices in $\Delta[n]$ with differential given as the alternating sum of face maps. Then $C: SAb \rightarrow Ch_{\geq 0}$ is the left Kan extension of $\Delta \rightarrow Ch_{\geq 0}$ and $\Gamma$ is the obvious right adjoint to that (it can't be anything else). (Once upon a time I learned this kind of construction from Maclane-Moerdijk, but it can also be found on the nLab :http://ncatlab.org/nlab/show/nerve+and+realization.)

One would then like to show that $C$ and $\Gamma$ are inverse equivalences by showing that the (obvious) unit $Id \rightarrow \Gamma C$ and counit $C\Gamma \rightarrow Id$ are isomorphisms.

By constrast, in every textbook treatment that I have seen, one constructs instead of a quotient complex $C$ an isomorphic subcomplex $N$, complementary to degenerate simplices, but you can do this in at least two different ways. Moreover, one constructs an isomorphism $\oplus_{[n] \rightarrow [k]} NA_{k} \rightarrow A_{n}$, where the sum is over surjections $[n] \rightarrow [k]$ in $\Delta$, and defines $\Gamma$ similarly, as $\Gamma(C)=\oplus_{[n] \rightarrow [k]} C_{k}$. Ultimately, it seems that one then shows $\Gamma N(A)\simeq A$ for $A \in SAb$ and $N\Gamma(C) \simeq C$ for $C \in Ch_{\geq 0}$, but the isomorphism $\Gamma N(A)\simeq A$ is somehow in the wrong direction (before you know it's an isomorphism), since a priori you would only expect $N,\Gamma$ to form an adjunction and the isomorphism should be given by the unit $A \simeq \Gamma N(A)$.

So my question is whether there is a 'more natural' treatment somewhere in the literature, phrased in terms of the quotient complex $C$, rather than the subcomplex $N$, and in terms of $\Gamma$ given by the obvious right adjoint, not as a sum over surjections, which at first seems to be pulled out of a hat. Of course one can unwind the usual proof and see that everything matches up. For example, one should be able to identify $Ch_{\geq 0}(C(\mathbf{Z}\Delta[n],C)$ with $\oplus_{[n]\rightarrow [k]} C_{k}$, but one can see at least two ways to do this.

If there is no such treatment, can one explain how to construct one or why one shouldn't bother?