The Stacks project

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. $\square$