Lemma 7.25.3 (03HU)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 7: Sites and Sheaves
Section 7.25: Localization
Lemma 7.25.3 (cite)

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Lemma 7.25.3. Let $\mathcal{C}$ be a site. Let $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ . Let $X/U$ be an object of $\mathcal{C}/U$ . Then we have $j_{U!}(h_{X/U}^\# ) = h_ X^\#$ .

Proof. Denote $p : X \to U$ the structure morphism of $X$ . By Lemma 7.25.2 we see $j_{U!}(h_{X/U}^\# )$ is the sheaf associated to the presheaf

$V \longmapsto \coprod \nolimits _{\varphi \in \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(V, U)} \{ \psi : V \to X \mid p \circ \psi = \varphi \}$

This is clearly the same thing as $\mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(V, X)$ . Hence the lemma follows. $\square$

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