Lemma 7.25.4. Let $\mathcal{C}$ be a site. Let $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$. The functor $j_{U!}$ gives an equivalence of categories

**Proof.**
Let us denote objects of $\mathcal{C}/U$ as pairs $(X, a)$ where $X$ is an object of $\mathcal{C}$ and $a : X \to U$ is a morphism of $\mathcal{C}$. Similarly, objects of $\mathop{\mathit{Sh}}\nolimits (\mathcal{C})/h_ U^\# $ are pairs $(\mathcal{F}, \varphi )$. The functor $\mathop{\mathit{Sh}}\nolimits (\mathcal{C}/U) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C})/h_ U^\# $ sends $\mathcal{G}$ to the pair $(j_{U!}\mathcal{G}, \gamma )$ where $\gamma $ is the composition of $j_{U!}\mathcal{G} \to j_{U!}*$ with the identification $j_{U!}* = h_ U^\# $.

Let us construct a functor from $\mathop{\mathit{Sh}}\nolimits (\mathcal{C})/h_ U^\# $ to $\mathop{\mathit{Sh}}\nolimits (\mathcal{C}/U)$. Suppose that $(\mathcal{F}, \varphi )$ is given. For an object $(X, a)$ of $\mathcal{C}/U$ we consider the set $\mathcal{F}_\varphi (X, a)$ of elements $s \in \mathcal{F}(X)$ which under $\varphi $ map to the image of $a \in \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(X, U) = h_ U(X)$ in $h_ U^\# (X)$. It is easy to see that $(X, a) \mapsto \mathcal{F}_\varphi (X, a)$ is a sheaf on $\mathcal{C}/U$. Clearly, the rule $(\mathcal{F}, \varphi ) \mapsto \mathcal{F}_\varphi $ defines a functor $\mathop{\mathit{Sh}}\nolimits (\mathcal{C})/h_ U^\# \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C}/U)$.

Consider also the functor $\textit{PSh}(\mathcal{C})/h_ U \to \textit{PSh}(\mathcal{C}/U)$, $(\mathcal{F}, \varphi ) \mapsto \mathcal{F}_\varphi $ where $\mathcal{F}_\varphi (X, a)$ is defined as the set of elements of $\mathcal{F}(X)$ mapping to $a \in h_ U(X)$. We claim that the diagram

commutes, where the vertical arrows are given by sheafification. To see this^{1}, it suffices to prove that the construction commutes with the functor $\mathcal{F} \mapsto \mathcal{F}^+$ of Lemmas 7.10.3 and 7.10.4 and Theorem 7.10.10. Commutation with $\mathcal{F} \mapsto \mathcal{F}^+$ follows from the fact that given $(X, a)$ the categories of coverings of $(X, a)$ in $\mathcal{C}/U$ and coverings of $X$ in $\mathcal{C}$ are canonically identified.

Next, let $\textit{PSh}(\mathcal{C}/U) \to \textit{PSh}(\mathcal{C})/h_ U$ send $\mathcal{G}$ to the pair $(j_{U!}^{PSh}\mathcal{G}, \gamma )$ where $j_{U!}^{PSh}\mathcal{G}$ the presheaf defined by the formula in Lemma 7.25.2 and $\gamma $ is the composition of $j_{U!}^{PSh}\mathcal{G} \to j_{U!}*$ with the identification $j_{U!}^{PSh}* = h_ U$ (obvious from the formula). Then it is immediately clear that the diagram

commutes, where the vertical arrows are sheafification. Putting everything together it suffices to show there are functorial isomorphisms $(j_{U!}^{PSh}\mathcal{G})_\gamma = \mathcal{G}$ for $\mathcal{G}$ in $\textit{PSh}(\mathcal{C}/U)$ and $j_{U!}^{PSh}\mathcal{F}_\varphi = \mathcal{F}$ for $(\mathcal{F}, \varphi )$ in $\textit{PSh}(\mathcal{C})/h_ U$. The value of the presheaf $(j_{U!}^{PSh}\mathcal{G})_\gamma $ on $(X, a)$ is the fibre of the map

over $a$ which is $\mathcal{G}(X, a)$. This proves the first equality. The value of the presheaf $j_{U!}^{PSh}\mathcal{F}_\varphi $ is on $X$ is

because given a set map $S \to S'$ the set $S$ is the disjoint union of its fibres. $\square$

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## Comments (2)

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