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 \ref{lemma-simplicial-site-site}). 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 \ref{lemma-simplicial-site-site}). 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. \medskip\noindent 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 \ref{lemma-simplicial-cocontinuous-site}). 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 \ref{lemma-simplicial-cocontinuous-site}). 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. \medskip\noindent At this point part (1) and the existence of $g^{Sh}_{n!}$ and $g_{n!}$ in cases A and B follows from Sites, Lemmas \ref{sites-lemma-cocontinuous-morphism-topoi} and \ref{sites-lemma-when-shriek} and Modules on Sites, Lemma \ref{sites-modules-lemma-g-shriek-adjoint}. \medskip\noindent 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 $$ \Mor_{\Sh(\mathcal{C}_n)}(\mathcal{G}, \mathcal{F}_n) = \Mor_{\Sh(\mathcal{C}_{total})}(\mathcal{H}, \mathcal{F}) $$ Thus $\mathcal{H} = g^{Sh}_{n!}\mathcal{G}$ as desired. \medskip\noindent 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 $$ \Mor_{\textit{Ab}(\mathcal{C}_n)}(\mathcal{G}, \mathcal{F}_n) = \Mor_{\textit{Ab}(\mathcal{C}_{total})}(\mathcal{H}, \mathcal{F}) $$ is proved exactly as above. Thus $\mathcal{H} = g_{n!}\mathcal{G}$ as desired. \medskip\noindent The exactness properties of $g^{Sh}_{n!}$ and $g_{n!}$ follow from formulas given for these functors.