I´m looking at the proof of the Dold-Kan correspondence. Let $SA$ be the category of simplicial objects in an abelian category $A$ and $CH_{\ge0}(A) $ the category of non-negative chain complexes in $A$. The theorem states that the normalisation functor $N : SA \rightarrow CH_{\ge0}(A) $ is an equivalence of categories.
I am following the proof in Weibel´s book and am trying to construct the inverse functor $K : CH_{\ge0}(A)\rightarrow SA$. For a chain complex $C$ we define $K(C)_n = \oplus_{p \le n} \oplus _{\eta} C_p[\eta ]$ where, for each $p$ $\eta$ ranges over all the surjections $[n]\rightarrow [p]$ and $C_p[\eta ]$ denotes a copy of $C_p$. I am trying to show that this is a simplicial object in our abelian category. I using the definition of a simplical object as a contravariant functor from the ordinal number category $\Delta \rightarrow A$. I understand how it is defined on objects and morphisms however I am stuck trying to show that $K(\alpha \circ \beta) = K(\beta) \circ K(\alpha)$ for $\alpha$, $\beta$ in $\Delta$.
Any help would be much appreciated.