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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