Lemma 21.9.1. The functor given by Equation (21.9.0.1) is an exact functor (see Homology, Lemma 12.7.2).

**Proof.**
For any object $W$ of $\mathcal{C}$ the functor $\mathcal{F} \mapsto \mathcal{F}(W)$ is an additive exact functor from $\textit{PAb}(\mathcal{C})$ to $\textit{Ab}$. The terms $\check{\mathcal{C}}^ p(\mathcal{U}, \mathcal{F})$ of the complex are products of these exact functors and hence exact. Moreover a sequence of complexes is exact if and only if the sequence of terms in a given degree is exact. Hence the lemma follows.
$\square$

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