The Stacks project

Definition 7.47.10. Let $\mathcal{C}$ be a category endowed with a topology $J$. Let $\mathcal{F}$ be a presheaf of sets on $\mathcal{C}$. We say that $\mathcal{F}$ is a sheaf on $\mathcal{C}$ if for every $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ and for every covering sieve $S$ of $U$ the canonical map

\[ \mathop{Mor}\nolimits _{\textit{PSh}(\mathcal{C})}(h_ U, \mathcal{F}) \longrightarrow \mathop{Mor}\nolimits _{\textit{PSh}(\mathcal{C})}(S, \mathcal{F}) \]

is bijective.

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