The Stacks project

Theorem 59.18.8. Notation and assumptions as in Definition 59.18.1. On $\textit{PAb}(\mathcal{C})$ the functors $\check{H}^ p(\mathcal{U}, -)$ are the right derived functors of $\check{H}^0(\mathcal{U}, -)$.

Proof. By the Lemma 59.18.7, the functors $\check H^ p(\mathcal{U}, -)$ are universal $\delta $-functors since they are effaceable. So are the right derived functors of $\check H^0(\mathcal{U}, -)$. Since they agree in degree $0$, they agree by the universal property of universal $\delta $-functors. For more details see Cohomology on Sites, Lemma 21.9.6. $\square$

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