The Stacks project

Remark 7.25.10. Localization and presheaves. Let $\mathcal{C}$ be a category. Let $U$ be an object of $\mathcal{C}$. Strictly speaking the functors $j_ U^{-1}$, $j_{U*}$ and $j_{U!}$ have not been defined for presheaves. But of course, we can think of a presheaf as a sheaf for the chaotic topology on $\mathcal{C}$ (see Example 7.6.6). Hence we also obtain a functor

\[ j_ U^{-1} : \textit{PSh}(\mathcal{C}) \longrightarrow \textit{PSh}(\mathcal{C}/U) \]

and functors

\[ j_{U*}, j_{U!} : \textit{PSh}(\mathcal{C}/U) \longrightarrow \textit{PSh}(\mathcal{C}) \]

which are right, left adjoint to $j_ U^{-1}$. By Lemma 7.25.2 we see that $j_{U!}\mathcal{G}$ is the presheaf

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

In addition the functor $j_{U!}$ commutes with fibre products and equalizers.

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