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 $\Sh(\mathcal{C})/h_U^\#$ are pairs $(\mathcal{F}, \varphi)$. The functor $\Sh(\mathcal{C}/U) \to \Sh(\mathcal{C})/h_U^\#$ 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 $j_{U!}* = h_U^\#$. \medskip\noindent Let us construct a functor from $\Sh(\mathcal{C})/h_U^\#$ to $\Sh(\mathcal{C}/U)$. 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 $a \in \Mor_\mathcal{C}(X, U) = h_U(X)$ 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 $\Sh(\mathcal{C})/h_U^\# \to \Sh(\mathcal{C}/U)$. \medskip\noindent 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 $$ \xymatrix{ \textit{PSh}(\mathcal{C})/h_U \ar[r] \ar[d] & \textit{PSh}(\mathcal{C}/U) \ar[d] \\ \Sh(\mathcal{C})/h_U^\# \ar[r] & \Sh(\mathcal{C}/U) } $$ commutes, where the vertical arrows are given by sheafification. To see this\footnote{An alternative is to describe $\mathcal{F}_\varphi$ by the cartesian diagram $$ \vcenter{ \xymatrix{ \mathcal{F}_\varphi \ar[r] \ar[d] & {*} \ar[d] \\ \mathcal{F}|_{\mathcal{C}/U} \ar[r] & h_U|_{\mathcal{C}/U} } } \quad\text{for presheaves and}\quad \vcenter{ \xymatrix{ \mathcal{F}_\varphi \ar[r] \ar[d] & {*} \ar[d] \\ \mathcal{F}|_{\mathcal{C}/U} \ar[r] & h_U^\#|_{\mathcal{C}/U} } } $$ for sheaves and use that restriction to $\mathcal{C}/U$ commutes with sheafification.}, it suffices to prove that the construction commutes with the functor $\mathcal{F} \mapsto \mathcal{F}^+$ of Lemmas \ref{lemma-plus-presheaf} and \ref{lemma-plus-functorial} and Theorem \ref{theorem-plus}. 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. \medskip\noindent 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 \ref{lemma-describe-j-shriek} 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 $$ \xymatrix{ \textit{PSh}(\mathcal{C}/U) \ar[r] \ar[d] & \textit{PSh}(\mathcal{C})/h_U \ar[d] \\ \Sh(\mathcal{C}/U) \ar[r] & \Sh(\mathcal{C})/h_U^\# } $$ 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 $$ \coprod\nolimits_{a' : X \to U} \mathcal{G}(X, a') \to \Mor_\mathcal{C}(X, U) $$ 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 $$ \coprod\nolimits_{a : X \to U} \mathcal{F}_\varphi(X, a) = \mathcal{F}(X) $$ because given a set map $S \to S'$ the set $S$ is the disjoint union of its fibres.