The Stacks project

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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