Let $A_\bullet$ be a (non-augmented) simplicial object in an abelian category, with face maps $d_i : A_n \to A_{n-1}$ and degeneracy maps $s_i : A_n \to A_{n+1}$, $0 \le i \le n$, for each $n \ge 0$. Let $C A_\bullet$ be the unnormalised chain complex associated with $A_\bullet$: so $C A_n = A_n$ (as objects) and the boundary operator $\partial : C A_n \to C A_{n-1}$ is defined by $\partial = \sum_{i=0}^{n} (-1)^i d_i$ Let $N A_\bullet$ be the normalised Moore complex, where $N A_n = \bigcap_{i=0}^{n-1} \ker (d_i : A_n \to A_{n-1})$ and the boundary operator $\partial : N A_n \to N A_{n-1}$ is the restriction of the face map $(-1)^n d_n : A_n \to A_{n-1}$. This is evidently a subcomplex of $C A_\bullet$, and there is a chain map $p : C A_\bullet \to N A_\bullet$ defined in degree $n$ by $p_n = q_{n-1} \circ \cdots \circ q_0$ where $q_i = \textrm{id} - s_i \circ d_i$. This is a retraction of the inclusion $N A_\bullet \hookrightarrow C A_\bullet$, and the Dold–Kan theorem tells us there is a short exact sequence of chain complexes $0 \to D A_\bullet \to C A_\bullet \overset{p}{\to} N A_\bullet \to 0$ where $D A_\bullet \to C A_\bullet$ is the inclusion of the subcomplex of degenerate simplices, defined by $D A_n = \sum_{i=0}^{n-1} \textrm{im}(s_i : A_{n-1} \to A_n)$ One then shows the induced maps $H_* (C A_\bullet) \to H_* (N A_\bullet)$ are isomorphisms by constructing appropriate chain homotopies. The long exact sequence of homology $\cdots \to H_{n+1} (D A_\bullet) \to H_n (C A_\bullet) \to H_n (N A_\bullet) \to H_n (D A_\bullet) \to \cdots$ then implies that $H_* (D A_\bullet) = 0$, as one expects.
Question. Can we show that $H_* (D A_\bullet) = 0$ more directly, by e.g. showing that $\textrm{id}_\bullet : D A_\bullet \to D A_\bullet$ is null-homotopic? I expect this should be true, since the corresponding simplicial subobject of $A_\bullet$ is contractible (when augmented with $0$). Goerss and Jardine give an explicit formula for the chain homotopy $\textrm{id}_A \Rightarrow p$ (see Ch. III, Thm 2.4), but an explicit formula for the chain homotopy $\textrm{id}_{D A} \Rightarrow 0$ has eluded me.