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{\mathrm{Mor}}\nolimits _{\textit{PSh}(\mathcal{C})}(h_ U, \mathcal{F}) \longrightarrow \mathop{\mathrm{Mor}}\nolimits _{\textit{PSh}(\mathcal{C})}(S, \mathcal{F})
is bijective.
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