Lemma 25.12.6. Let $\mathcal{C}$ be a site. Let $\mathcal{B} \subset \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ be a subset. Assume

$\mathcal{C}$ has fibre products,

for all $X \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ there exists a finite covering $\{ U_ i \to X\} _{i \in I}$ with $U_ i \in \mathcal{B}$,

if $\{ U_ i \to X\} _{i \in I}$ is a finite covering with $U_ i \in \mathcal{B}$ and $U \to X$ is a morphism with $U \in \mathcal{B}$, then $\{ U_ i \to X\} _{i \in I} \amalg \{ U \to X\} $ is a covering.

Then for every $X$ there exists a hypercovering $K$ of $X$ such that each $K_ n = \{ U_{n, i} \to X\} _{i \in I_ n}$ with $I_ n$ finite and $U_{n, i} \in \mathcal{B}$.

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