Assume $u$ satisfies properties (1) -- (5). We will show that the adjunction mappings $$ \mathcal{G} \longrightarrow g_*g^{-1}\mathcal{G} \quad\text{and}\quad g^{-1}g_*\mathcal{F} \longrightarrow \mathcal{F} $$ are isomorphisms. \medskip\noindent Note that Lemma \ref{lemma-when-shriek} applies and we have $g^{-1}\mathcal{G}(U) = \mathcal{G}(u(U))$ for any sheaf $\mathcal{G}$ on $\mathcal{D}$. Next, let $\mathcal{F}$ be a sheaf on $\mathcal{C}$, and let $V$ be an object of $\mathcal{D}$. By definition we have $g_*\mathcal{F}(V) = \lim_{u(U) \to V} \mathcal{F}(U)$. Hence $$ g^{-1}g_*\mathcal{F}(U) = \lim_{U', u(U') \to u(U)} \mathcal{F}(U') $$ where the morphisms $\psi : u(U') \to u(U)$ need not be of the form $u(\alpha)$. The category of such pairs $(U', \psi)$ has a final object, namely $(U, \text{id})$, which gives rise to the map from the limit into $\mathcal{F}(U)$. Let $(s_{(U', \psi)})$ be an element of the limit. We want to show that $s_{(U', \psi)}$ is uniquely determined by the value $s_{(U, \text{id})} \in \mathcal{F}(U)$. By property (4) given any $(U', \psi)$ there exists a covering $\{U'_i \to U'\}$ such that the compositions $u(U'_i) \to u(U') \to u(U)$ are of the form $u(c_i)$ for some $c_i : U'_i \to U$ in $\mathcal{C}$. Hence $$ s_{(U', \psi)}|_{U'_i} = c_i^*(s_{(U, \text{id})}). $$ Since $\mathcal{F}$ is a sheaf it follows that indeed $s_{(U', \psi)}$ is determined by $s_{(U, \text{id})}$. This proves uniqueness. For existence, assume given any $s \in \mathcal{F}(U)$, $\psi : u(U') \to u(U)$, $\{f_i : U_i' \to U'\}$ and $c_i : U_i' \to U$ such that $\psi \circ u(f_i) = u(c_i)$ as above. We claim there exists a (unique) element $s_{(U', \psi)} \in \mathcal{F}(U')$ such that $$ s_{(U', \psi)}|_{U'_i} = c_i^*(s). $$ Namely, a priori it is not clear the elements $c_i^*(s)|_{U_i' \times_{U'} U_j'}$ and $c_j^*(s)|_{U_i' \times_{U'} U_j'}$ agree, since the diagram $$ \xymatrix{ U_i' \times_{U'} U_j' \ar[r]_-{\text{pr}_2} \ar[d]_{\text{pr}_1} & U_j' \ar[d]^{c_j} \\ U_i' \ar[r]^{c_i} & U} $$ need not commute. But condition (3) of the lemma guarantees that there exist coverings $\{f_{ijk} : U'_{ijk} \to U_i' \times_{U'} U_j'\}_{k \in K_{ij}}$ such that $c_i \circ \text{pr}_1 \circ f_{ijk} = c_j \circ \text{pr}_2 \circ f_{ijk}$. Hence $$ f_{ijk}^* \left(c_i^*s|_{U_i' \times_{U'} U_j'}\right) = f_{ijk}^* \left(c_j^*s|_{U_i' \times_{U'} U_j'}\right) $$ Hence $c_i^*(s)|_{U_i' \times_{U'} U_j'} = c_j^*(s)|_{U_i' \times_{U'} U_j'}$ by the sheaf condition for $\mathcal{F}$ and hence the existence of $s_{(U', \psi)}$ also by the sheaf condition for $\mathcal{F}$. The uniqueness guarantees that the collection $(s_{(U', \psi)})$ so obtained is an element of the limit with $s_{(U, \psi)} = s$. This proves that $g^{-1}g_*\mathcal{F} \to \mathcal{F}$ is an isomorphism. \medskip\noindent Let $\mathcal{G}$ be a sheaf on $\mathcal{D}$. Let $V$ be an object of $\mathcal{D}$. Then we see that $$ g_*g^{-1}\mathcal{G}(V) = \lim_{U, \psi : u(U) \to V} \mathcal{G}(u(U)) $$ By the preceding paragraph we see that the value of the sheaf $g_*g^{-1}\mathcal{G}$ on an object $V$ of the form $V = u(U)$ is equal to $\mathcal{G}(u(U))$. (Formally, this holds because we have $g^{-1}g_*g^{-1} \cong g^{-1}$, and the description of $g^{-1}$ given at the beginning of the proof; informally just by comparing limits here and above.) Hence the adjunction mapping $\mathcal{G} \to g_*g^{-1}\mathcal{G}$ has the property that it is a bijection on sections over any object of the form $u(U)$. Since by axiom (5) there exists a covering of $V$ by objects of the form $u(U)$ we see easily that the adjunction map is an isomorphism.