**Proof.**
Case A. If $\{ U_ i \to U\} _{i \in I}$ is a covering in $\mathcal{C}_ n$ then the image $\{ U_ i \to U\} _{i \in I}$ is a covering in $\mathcal{C}_{total}$ by definition (Lemma 81.3.1). For a morphism $V \to U$ of $\mathcal{C}_ n$, the fibre product $V \times _ U U_ i$ in $\mathcal{C}_ n$ is also the fibre product in $\mathcal{C}_{total}$ (by the claim in the proof of Lemma 81.3.1). Therefore our functor is continuous. On the other hand, our functor defines a bijection between coverings of $U$ in $\mathcal{C}_ n$ and coverings of $U$ in $\mathcal{C}_{total}$. Therefore it is certainly the case that our functor is cocontinuous.

Case B. If $\{ U_ i \to U\} _{i \in I}$ is a covering in $\mathcal{C}_ n$ then the image $\{ U_ i \to U\} _{i \in I}$ is a covering in $\mathcal{C}_{total}$ by definition (Lemma 81.3.2). For a morphism $V \to U$ of $\mathcal{C}_ n$, the fibre product $V \times _ U U_ i$ in $\mathcal{C}_ n$ is also the fibre product in $\mathcal{C}_{total}$ (by the claim in the proof of Lemma 81.3.2). Therefore our functor is continuous. On the other hand, our functor defines a bijection between coverings of $U$ in $\mathcal{C}_ n$ and coverings of $U$ in $\mathcal{C}_{total}$. Therefore it is certainly the case that our functor is cocontinuous.

At this point part (1) and the existence of $g^{Sh}_{n!}$ and $g_{n!}$ in cases A and B follows from Sites, Lemmas 7.21.1 and 7.21.5 and Modules on Sites, Lemma 18.16.2.

Proof of (3). Let $\mathcal{G}$ be a sheaf on $\mathcal{C}_ n$. Consider the sheaf $\mathcal{H}$ on $\mathcal{C}_{total}$ whose degree $m$ part is the sheaf

\[ \mathcal{H}_ m = \coprod \nolimits _{\varphi : [n] \to [m]} f_\varphi ^{-1}\mathcal{G} \]

given in part (3) of the statement of the lemma. Given a map $\psi : [m] \to [m']$ the map $\mathcal{H}(\psi ) : f_\psi ^{-1}\mathcal{H}_ m \to \mathcal{H}_{m'}$ is given on components by the identifications

\[ f_\psi ^{-1} f_\varphi ^{-1} \mathcal{G} \to f_{\psi \circ \varphi }^{-1}\mathcal{G} \]

Observe that given a map $\alpha : \mathcal{H} \to \mathcal{F}$ of sheaves on $\mathcal{C}_{total}$ we obtain a map $\mathcal{G} \to \mathcal{F}_ n$ corresponding to the restriction of $\alpha _ n$ to the component $\mathcal{G}$ in $\mathcal{H}_ n$. Conversely, given a map $\beta : \mathcal{G} \to \mathcal{F}_ n$ of sheaves on $\mathcal{C}_ n$ we can define $\alpha : \mathcal{H} \to \mathcal{F}$ by letting $\alpha _ m$ be the map which on components

\[ f_\varphi ^{-1}\mathcal{G} \to \mathcal{F}_ m \]

uses the maps adjoint to $\mathcal{F}(\varphi ) \circ f_\varphi ^{-1}\beta $. We omit the arguments showing these two constructions give mutually inverse maps

\[ \mathop{Mor}\nolimits _{\mathop{\mathit{Sh}}\nolimits (\mathcal{C}_ n)}(\mathcal{G}, \mathcal{F}_ n) = \mathop{Mor}\nolimits _{\mathop{\mathit{Sh}}\nolimits (\mathcal{C}_{total})}(\mathcal{H}, \mathcal{F}) \]

Thus $\mathcal{H} = g^{Sh}_{n!}\mathcal{G}$ as desired.

Proof of (4). If $\mathcal{G}$ is an abelian sheaf on $\mathcal{C}_ n$, then we proceed in exactly the same ammner as above, except that we define $\mathcal{H}$ is the abelian sheaf on $\mathcal{C}_{total}$ whose degree $m$ part is the sheaf

\[ \bigoplus \nolimits _{\varphi : [n] \to [m]} f_\varphi ^{-1}\mathcal{G} \]

with transition maps defined exactly as above. The bijection

\[ \mathop{Mor}\nolimits _{\textit{Ab}(\mathcal{C}_ n)}(\mathcal{G}, \mathcal{F}_ n) = \mathop{Mor}\nolimits _{\textit{Ab}(\mathcal{C}_{total})}(\mathcal{H}, \mathcal{F}) \]

is proved exactly as above. Thus $\mathcal{H} = g_{n!}\mathcal{G}$ as desired.

The exactness properties of $g^{Sh}_{n!}$ and $g_{n!}$ follow from formulas given for these functors.
$\square$

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