Lemma 59.18.6. The complex of abelian presheaves
\begin{align*} \mathbf{Z}_\mathcal {U}^\bullet \quad : \quad \bigoplus _{i_0 \in I} \mathbf{Z}_{U_{i_0}} \leftarrow \bigoplus _{i_0, i_1 \in I} \mathbf{Z}_{U_{i_0} \times _ U U_{i_1}} \leftarrow \bigoplus _{i_0, i_1, i_2 \in I} \mathbf{Z}_{U_{i_0} \times _ U U_{i_1} \times _ U U_{i_2}} \leftarrow \ldots \end{align*}
is exact in all degrees except $0$ in $\textit{PAb}(\mathcal{C})$.
Proof.
For any $V\in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ the complex of abelian groups $\mathbf{Z}_\mathcal {U}^\bullet (V)$ is
\[ \begin{matrix} \mathbf{Z}\left[ \coprod _{i_0\in I} \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(V, U_{i_0})\right] \leftarrow \mathbf{Z}\left[ \coprod _{i_0, i_1 \in I} \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(V, U_{i_0} \times _ U U_{i_1})\right] \leftarrow \ldots =
\\ \bigoplus _{\varphi : V \to U} \left( \mathbf{Z}\left[\coprod _{i_0 \in I} \mathop{\mathrm{Mor}}\nolimits _\varphi (V, U_{i_0})\right] \leftarrow \mathbf{Z}\left[\coprod _{i_0, i_1\in I} \mathop{\mathrm{Mor}}\nolimits _\varphi (V, U_{i_0}) \times \mathop{\mathrm{Mor}}\nolimits _\varphi (V, U_{i_1})\right] \leftarrow \ldots \right)
\end{matrix} \]
where
\[ \mathop{\mathrm{Mor}}\nolimits _{\varphi }(V, U_ i) = \{ V \to U_ i \text{ such that } V \to U_ i \to U \text{ equals } \varphi \} . \]
Set $S_\varphi = \coprod _{i\in I} \mathop{\mathrm{Mor}}\nolimits _\varphi (V, U_ i)$, so that
\[ \mathbf{Z}_\mathcal {U}^\bullet (V) = \bigoplus _{\varphi : V \to U} \left( \mathbf{Z}[S_\varphi ] \leftarrow \mathbf{Z}[S_\varphi \times S_\varphi ] \leftarrow \mathbf{Z}[S_\varphi \times S_\varphi \times S_\varphi ] \leftarrow \ldots \right). \]
Thus it suffices to show that for each $S = S_\varphi $, the complex
\begin{align*} \mathbf{Z}[S] \leftarrow \mathbf{Z}[S \times S] \leftarrow \mathbf{Z}[S \times S \times S] \leftarrow \ldots \end{align*}
is exact in negative degrees. To see this, we can give an explicit homotopy. Fix $s\in S$ and define $K: n_{(s_0, \ldots , s_ p)} \mapsto n_{(s, s_0, \ldots , s_ p)}.$ One easily checks that $K$ is a nullhomotopy for the operator
\[ \delta : \eta _{(s_0, \ldots , s_ p)} \mapsto \sum \nolimits _{i = 0}^ p (-1)^ p \eta _{(s_0, \ldots , \hat s_ i, \ldots , s_ p)}. \]
See Cohomology on Sites, Lemma 21.9.4 for more details.
$\square$
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