Definition 7.7.1. Let $\mathcal{C}$ be a site, and let $\mathcal{F}$ be a presheaf of sets on $\mathcal{C}$. We say $\mathcal{F}$ is a sheaf if for every covering $\{ U_ i \to U\} _{i \in I} \in \text{Cov}(\mathcal{C})$ the diagram

7.7.1.1
$$\label{sites-equation-sheaf-condition} \xymatrix{ \mathcal{F}(U) \ar[r] & \prod \nolimits _{i\in I} \mathcal{F}(U_ i) \ar@<1ex>[r]^-{\text{pr}_0^*} \ar@<-1ex>[r]_-{\text{pr}_1^*} & \prod \nolimits _{(i_0, i_1) \in I \times I} \mathcal{F}(U_{i_0} \times _ U U_{i_1}) }$$

represents the first arrow as the equalizer of $\text{pr}_0^*$ and $\text{pr}_1^*$.

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