The Stacks project

Lemma 7.25.2. Let $\mathcal{C}$ be a site. Let $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$. Let $\mathcal{G}$ be a presheaf on $\mathcal{C}/U$. Then $j_{U!}(\mathcal{G}^\# )$ is the sheaf associated to the presheaf

\[ V \longmapsto \coprod \nolimits _{\varphi \in \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(V, U)} \mathcal{G}(V \xrightarrow {\varphi } U) \]

with obvious restriction mappings.

Proof. By Lemma 7.21.5 we have $j_{U!}(\mathcal{G}^\# ) = ((j_ U)_ p\mathcal{G}^\# )^\# $. By Lemma 7.13.4 this is equal to $((j_ U)_ p\mathcal{G})^\# $. Hence it suffices to prove that $(j_ U)_ p$ is given by the formula above for any presheaf $\mathcal{G}$ on $\mathcal{C}/U$. OK, and by the definition in Section 7.5 we have

\[ (j_ U)_ p\mathcal{G}(V) = \mathop{\mathrm{colim}}\nolimits _{(W/U, V \to W)} \mathcal{G}(W) \]

Now it is clear that the category of pairs $(W/U, V \to W)$ has an object $O_\varphi = (\varphi : V \to U, \text{id} : V \to V)$ for every $\varphi : V \to U$, and moreover for any object there is a unique morphism from one of the $O_\varphi $ into it. The result follows. $\square$

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