The Stacks project

Proof. First note that the map $\mathcal{F}(U) \to \prod \nolimits _{x \in U} \mathcal{F}_ x$ is injective by the uniqueness in the sheaf condition of Definition 6.30.2. Let $(s_ x)$ be any element on the right hand side which satisfies $(*)$. Clearly this means we can find a covering $U = \bigcup U_ i$, $U_ i \in \mathcal{B}$ such that $(s_ x)_{x \in U_ i}$ comes from certain $\sigma _ i \in \mathcal{F}(U_ i)$. For every $y \in U_ i \cap U_ j$ the sections $\sigma _ i$ and $\sigma _ j$ agree in the stalk $\mathcal{F}_ y$. Hence there exists an element $V_{ijy} \in \mathcal{B}$, $y \in V_{ijy}$ such that $\sigma _ i|_{V_{ijy}} = \sigma _ j|_{V_{ijy}}$. Thus the sheaf condition $(**)$ of Definition 6.30.2 applies to the system of $\sigma _ i$ and we obtain a section $s \in \mathcal{F}(U)$ with the desired property. $\square$