The Stacks project

Lemma 7.10.15. Let $\mathcal{C}$ be a site. Let $\mathcal{F}$ be a presheaf of sets on $\mathcal{C}$. Denote $\theta ^2 : \mathcal{F} \to \mathcal{F}^\#$ the canonical map of $\mathcal{F}$ into its sheafification. Let $U$ be an object of $\mathcal{C}$. Let $s \in \mathcal{F}^\# (U)$. There exists a covering $\{ U_ i \to U\}$ and sections $s_ i \in \mathcal{F}(U_ i)$ such that

1. $s|_{U_ i} = \theta ^2(s_ i)$, and

2. for every $i, j$ there exists a covering $\{ U_{ijk} \to U_ i \times _ U U_ j\}$ of $\mathcal{C}$ such that the pullbacks of $s_ i$ and $s_ j$ to each $U_{ijk}$ agree.

Conversely, given any covering $\{ U_ i \to U\}$, elements $s_ i \in \mathcal{F}(U_ i)$ such that (2) holds, then there exists a unique section $s \in \mathcal{F}^\# (U)$ such that (1) holds.