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 this1, 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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