The Stacks project

Remark 59.55.6. In the situation of Lemma 59.55.5 if $\mathcal{G}$ is a sheaf of sets on $Y_{\acute{e}tale}$, then we have

\[ \Gamma (Y, \mathcal{G}) = \text{Equalizer}( \xymatrix{ \Gamma (X_0, f_0^{-1}\mathcal{G}) \ar@<1ex>[r] \ar@<-1ex>[r] & \Gamma (X_1, f_1^{-1}\mathcal{G}) } ) \]

This is proved in exactly the same way, by showing that the sheaf $\mathcal{G}$ is the equalizer of the two maps $f_{0, *}f_0^{-1}\mathcal{G} \to f_{1, *}f_1^{-1}\mathcal{G}$.

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