It is immediate from the arguments in the proof of Lemma \ref{lemma-colimit-sites} that the functors $u_i$ are continuous. To finish the proof we have to show that $f_i^{-1} := u_{i, s}$ is an exact functor $\Sh(\mathcal{C}_i) \to \Sh(\mathcal{C})$. In fact it suffices to show that $f_i^{-1}$ is left exact, because it is right exact as a left adjoint (Categories, Lemma \ref{categories-lemma-exact-adjoint}). We first prove (\ref{equation-compute-pullback-to-limit}) and then we deduce exactness. \medskip\noindent For an arbitrary object $V$ of $\mathcal{C}$ we can pick a $a : j \to i$ and an object $V_j \in \Ob(\mathcal{C})$ with $V = u_j(V_j)$. Then we can set $$ \mathcal{G}(V) = \colim_{b : k \to j} f_{a \circ b}^{-1}\mathcal{F}(u_b(V_j)) $$ The value $\mathcal{G}(V)$ of the colimit is independent of the choice of $b : j \to i$ and of the object $V_j$ with $u_j(V_j) = V$; we omit the verification. Moreover, if $\alpha : V \to V'$ is a morphism of $\mathcal{C}$, then we can choose $b : j \to i$ and a morphism $\alpha_j : V_j \to V'_j$ with $u_j(\alpha_j) = \alpha$. This induces a map $\mathcal{G}(V') \to \mathcal{G}(V)$ by using the restrictions along the morphisms $u_b(\alpha_j) : u_b(V_j) \to u_b(V'_j)$. A check shows that $\mathcal{G}$ is a presheaf (omitted). In fact, $\mathcal{G}$ satisfies the sheaf condition. Namely, any covering $\mathcal{U} = \{U^t \to U\}$ in $\mathcal{C}$ comes from a finite level. Say $\mathcal{U}_j = \{U^t_j \to U_j\}$ is mapped to $\mathcal{U}$ by $u_j$ for some $a : j \to i$ in $\mathcal{I}$. Then we have $$ H^0(\mathcal{U}, \mathcal{G}) = \colim_{b : k \to j} H^0(u_b(\mathcal{U}_j), f_{b \circ a}^{-1}\mathcal{F}) = \colim_{b : k \to j} f_{b \circ a}^{-1}\mathcal{F}(u_b(U_j)) = \mathcal{G}(U) $$ as desired. The first equality holds because filtered colimits commute with finite limits (Categories, Lemma \ref{categories-lemma-directed-commutes}). By construction $\mathcal{G}(U)$ is given by the right hand side of (\ref{equation-compute-pullback-to-limit}). Hence (\ref{equation-compute-pullback-to-limit}) is true if we can show that $\mathcal{G}$ is equal to $f_i^{-1}\mathcal{F}$. \medskip\noindent In this paragraph we check that $\mathcal{G}$ is canonically isomorphic to $f_i^{-1}\mathcal{F}$. We strongly encourage the reader to skip this paragraph. To check this we have to show there is a bijection $\Mor_{\Sh(\mathcal{C})}(\mathcal{G}, \mathcal{H}) = \Mor_{\Sh(\mathcal{C}_i)}(\mathcal{F}, f_{i, *}\mathcal{H})$ functorial in the sheaf $\mathcal{H}$ on $\mathcal{C}$ where $f_{i, *} = u_i^p$. A map $\mathcal{G} \to \mathcal{H}$ is the same thing as a compatible system of maps $$ \varphi_{a, b, V_j} : f_{a \circ b}^{-1}\mathcal{F}(u_b(V_j)) \longrightarrow \mathcal{H}(u_j(V_j)) $$ for all $a : j \to i$, $b : k \to j$ and $V_j \in \Ob(\mathcal{C}_j)$. The compatibilities force the maps $\varphi_{a, b, V_j}$ to be equal to $\varphi_{a \circ b, \text{id}, u_b(V_j)}$. Given $a : j \to i$, the family of maps $\varphi_{a, \text{id}, V_j}$ corresponds to a map of sheaves $\varphi_a : f_a^{-1}\mathcal{F} \to f_{j, *}\mathcal{H}$. The compatibilities between the $\varphi_{a, \text{id}, u_a(V_i)}$ and the $\varphi_{\text{id}, \text{id}, V_i}$ implies that $\varphi_a$ is the adjoint of the map $\varphi_{id}$ via $$ \Mor_{\Sh(\mathcal{C}_j)}(f_a^{-1}\mathcal{F}, f_{j, *}\mathcal{H}) = \Mor_{\Sh(\mathcal{C}_i)}(\mathcal{F}, f_{a, *}f_{j, *}\mathcal{H}) = \Mor_{\Sh(\mathcal{C}_i)}(\mathcal{F}, f_{i, *}\mathcal{H}) $$ Thus finally we see that the whole system of maps $\varphi_{a, b, V_j}$ is determined by the map $\varphi_{id} : \mathcal{F} \to f_{i, *}\mathcal{H}$. Conversely, given such a map $\psi : \mathcal{F} \to f_{i, *}\mathcal{H}$ we can read the argument just given backwards to construct the family of maps $\varphi_{a, b, V_j}$. This finishes the proof that $\mathcal{G} = f_i^{-1}\mathcal{F}$. \medskip\noindent Assume (\ref{equation-compute-pullback-to-limit}) holds. Then the functor $\mathcal{F} \mapsto f_i^{-1}\mathcal{F}(U)$ commutes with finite limits because finite limits of sheaves are computed in the category of presheaves (Lemma \ref{lemma-limit-sheaf}), the functors $f_a^{-1}$ commutes with finite limits, and filtered colimits commute with finite limits. To see that $\mathcal{F} \mapsto f_i^{-1}\mathcal{F}(V)$ commutes with finite limits for a general object $V$ of $\mathcal{C}$, we can use the same argument using the formula for $f_i^{-1}\mathcal{F}(V) = \mathcal{G}(V)$ given above. Thus $f_i^{-1}$ is left exact and the proof of the lemma is complete.