Remark 21.39.4. Notation and assumptions as in Example 21.39.1. Let $\mathcal{F}^\bullet$ be a bounded complex of abelian sheaves on $\mathcal{C}$. For any object $U$ of $\mathcal{C}$ there is a canonical map

$\mathcal{F}^\bullet (U) \longrightarrow L\pi _!(\mathcal{F}^\bullet )$

in $D(\textit{Ab})$. If $\mathcal{F}^\bullet$ is a complex of $\underline{B}$-modules then this map is in $D(B)$. To prove this, note that we compute $L\pi _!(\mathcal{F}^\bullet )$ by taking a quasi-isomorphism $\mathcal{P}^\bullet \to \mathcal{F}^\bullet$ where $\mathcal{P}^\bullet$ is a complex of projectives. However, since the topology is chaotic this means that $\mathcal{P}^\bullet (U) \to \mathcal{F}^\bullet (U)$ is a quasi-isomorphism hence can be inverted in $D(\textit{Ab})$, resp. $D(B)$. Composing with the canonical map $\mathcal{P}^\bullet (U) \to \pi _!(\mathcal{P}^\bullet )$ coming from the computation of $\pi _!$ as a colimit we obtain the desired arrow.

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