Proof.
It is immediate from the arguments in the proof of Lemma 7.18.2 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 $\mathop{\mathit{Sh}}\nolimits (\mathcal{C}_ i) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C})$. In fact it suffices to show that $f_ i^{-1}$ is left exact, because it is right exact as a left adjoint (Categories, Lemma 4.24.6). We first prove (7.18.3.1) and then we deduce exactness.
For an arbitrary object $V$ of $\mathcal{C}$ we can pick a $a : j \to i$ and an object $V_ j \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ with $V = u_ j(V_ j)$. Then we can set
\[ \mathcal{G}(V) = \mathop{\mathrm{colim}}\nolimits _{b : k \to j} f_{a \circ b}^{-1}\mathcal{F}(u_ b(V_ j)) \]
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
\[ H^0(\mathcal{U}, \mathcal{G}) = \mathop{\mathrm{colim}}\nolimits _{b : k \to j} H^0(u_ b(\mathcal{U}_ j), f_{b \circ a}^{-1}\mathcal{F}) = \mathop{\mathrm{colim}}\nolimits _{b : k \to j} f_{b \circ a}^{-1}\mathcal{F}(u_ b(U_ j)) = \mathcal{G}(U) \]
as desired. The first equality holds because filtered colimits commute with finite limits (Categories, Lemma 4.19.2). By construction $\mathcal{G}(U)$ is given by the right hand side of (7.18.3.1). Hence (7.18.3.1) is true if we can show that $\mathcal{G}$ is equal to $f_ i^{-1}\mathcal{F}$.
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 $\mathop{\mathrm{Mor}}\nolimits _{\mathop{\mathit{Sh}}\nolimits (\mathcal{C})}(\mathcal{G}, \mathcal{H}) = \mathop{\mathrm{Mor}}\nolimits _{\mathop{\mathit{Sh}}\nolimits (\mathcal{C}_ i)}(\mathcal{F}, f_{i, *}\mathcal{H})$ 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 $V_ j \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C}_ j)$. 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
\[ \mathop{\mathrm{Mor}}\nolimits _{\mathop{\mathit{Sh}}\nolimits (\mathcal{C}_ j)}(f_ a^{-1}\mathcal{F}, f_{j, *}\mathcal{H}) = \mathop{\mathrm{Mor}}\nolimits _{\mathop{\mathit{Sh}}\nolimits (\mathcal{C}_ i)}(\mathcal{F}, f_{a, *}f_{j, *}\mathcal{H}) = \mathop{\mathrm{Mor}}\nolimits _{\mathop{\mathit{Sh}}\nolimits (\mathcal{C}_ i)}(\mathcal{F}, f_{i, *}\mathcal{H}) \]
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 $\mathcal{G} = f_ i^{-1}\mathcal{F}$.
Assume (7.18.3.1) 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 7.10.1), 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.
$\square$
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