The Stacks project

Situation 21.25.5. Let $f : (\mathcal{C}, \mathcal{O}) \to (\mathcal{C}', \mathcal{O}')$ be a morphism of ringed sites. Let $u : \mathcal{C}' \to \mathcal{C}$ be the corresponding continuous functor of sites. Let $\mathcal{A} \subset \textit{Mod}(\mathcal{O})$ be a weak Serre subcategory. We assume the following is true: there exists a subset $\mathcal{B}' \subset \mathop{\mathrm{Ob}}\nolimits (\mathcal{C}')$ such that

1. every object of $\mathcal{C}'$ has a covering whose members are in $\mathcal{B}'$, and

2. for every $V' \in \mathcal{B}'$ there exists an integer $d_{V'}$ and a cofinal system $\text{Cov}_{V'}$ of coverings of $V'$ such that

   $$H^ p(u(V'_ i), \mathcal{F}) = 0 \text{ for } \{ V'_ i \to V'\} \in \text{Cov}_{V'},\ p > d_{V'}, \text{ and } \mathcal{F} \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{A})$$