Lemma 7.45.2 (0792)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 7: Sites and Sheaves
Section 7.45: Pullback maps
Lemma 7.45.2 (cite)

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Lemma 7.45.2. Let $\mathcal{C}$ be a site. Let $a, b : V \to U$ be objects of $\mathcal{C}$ such that

$\xymatrix{ h_ V^\# \ar@<1ex>[r] \ar@<-1ex>[r] & h_ U^\# \ar[r] & {*} }$

is a coequalizer in $\mathop{\mathit{Sh}}\nolimits (\mathcal{C})$ . Then $\Gamma (\mathcal{C}, \mathcal{F})$ is the equalizer of $a^*, b^* : \mathcal{F}(U) \to \mathcal{F}(V)$ .

Proof. Since $\mathop{\mathrm{Mor}}\nolimits _{\mathop{\mathit{Sh}}\nolimits (\mathcal{C})}(h_ U^\# , \mathcal{F}) = \mathcal{F}(U)$ this is clear from the definitions. $\square$

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