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.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).