I am trying to work out exercise 8.4.2 from Weibel's Simplicial Methods in Homological Algebra. That is, I want to show that the Dold Kan equivalence in fact is an adjoint equivalence. I wanted to check whether my reasoning is correct or not.
The relevant functors are summarized the following. The normalized Moore functor $N:\mathcal{sAb}\to \mathcal{Ch}_{+}$ mapping a simplicial abelian group $A_\bullet$ to the chain complex with in degree $n$ the quotient $A_n/DA_n$, where $DA_n$ is given by the degenerate $n$-simplices in $A_\bullet$. Its inverse functor $\Gamma$ maps a chain complex $C$ to the simplicial abelian group which in degree $n$ is given by $$\bigoplus_{n\twoheadrightarrow k} C_k.$$
The natural isomorphism $\eta:id \to N\Gamma $ on component $C$ is the chain complex morphism which in degree $n$ is defined by the following composition $$C_n \hookrightarrow \bigoplus_{n\twoheadrightarrow k} C_k \to \bigoplus_{n\twoheadrightarrow k} C_k/\bigoplus_{n\twoheadrightarrow k, k\neq n} C_k\to C_n $$where the last map is an isomorphism. This map is an isomorphism on $C_n$, however it seems to me that I cannot say that this map really is the identity.
The map $\epsilon:\Gamma N\to id$ is given on component $A_\bullet$ by first using that $(NA_\bullet)_n$ is isomorphic to the intersection $$\bigcap_{i=0}^{n-1} \ker(d_i)\subset A_n$$ and then mapping $x_\sigma$ in $\bigoplus_{n\twoheadrightarrow k} C_k$, where $\sigma$ denotes a surjection $[n]\to [k]$ to the $\sigma^*(x)$ in $A_n$.
To show that these two natural transformations give $N$ and $\Gamma$ the structure of an adjoint, we want to show that the triangle identities $\epsilon \Gamma \circ \Gamma \eta=id$ and $N\epsilon \circ \eta N=id$ are satisfied. However, I cannot see why this should necessarily hold. There are many isomorphisms involved in the construction, and although it is clear that these identities hold up to some canonical isomorphism, I dont see why they should exactly be the same. Perhaps I'm overthinking things, or perhaps I'm missing out on some part of the theory.
Thanks in advance!