# The Stacks Project

## Tag 0494

Remark 7.24.9. 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 7.24.2 we see that $j_{U!}\mathcal{G}$ is the presheaf $$V \longmapsto \coprod\nolimits_{\varphi \in \mathop{\rm Mor}\nolimits_\mathcal{C}(V, U)} \mathcal{G}(V \xrightarrow{\varphi} U)$$ In addition the functor $j_{U!}$ commutes with fibre products and equalizers.

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\begin{remark}
\label{remark-localize-presheaves}
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 \ref{example-indiscrete}).
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 \ref{lemma-describe-j-shriek}
we see that $j_{U!}\mathcal{G}$ is the presheaf
$$V \longmapsto \coprod\nolimits_{\varphi \in \Mor_\mathcal{C}(V, U)} \mathcal{G}(V \xrightarrow{\varphi} U)$$
In addition the functor $j_{U!}$ commutes with fibre products and
equalizers.
\end{remark}

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