The Stacks project

Proof. By Lemma 59.73.10 the functor in each case is fully faithful. By the same lemma, all we have to show to finish the proof in case (1) is the following: given a constructible sheaf $\mathcal{F}_ i$ on $X_ i$ whose pullback $\mathcal{F}$ to $X$ is finite locally constant, there exists an $i' \geq i$ such that the pullback $\mathcal{F}_{i'}$ of $\mathcal{F}_ i$ to $X_{i'}$ is finite locally constant. By assumption there exists an étale covering $\mathcal{U} = \{ U_ j \to X\} _{j \in J}$ such that $\mathcal{F}|_{U_ j} \cong \underline{S_ j}$ for some finite set $S_ j$. We may assume $U_ j$ is affine for all $j \in J$. Since $X$ is quasi-compact, we may assume $J$ finite. By Lemma 59.51.2 we can find an $i' \geq i$ and an étale covering $\mathcal{U}_{i'} = \{ U_{i', j} \to X_{i'}\} _{j \in J}$ whose base change to $X$ is $\mathcal{U}$. Then $\mathcal{F}_{i'}|_{U_{i', j}}$ and $\underline{S_ j}$ are constructible sheaves on $(U_{i', j})_{\acute{e}tale}$ whose pullbacks to $U_ j$ are isomorphic. Hence after increasing $i'$ we get that $\mathcal{F}_{i'}|_{U_{i', j}}$ and $\underline{S_ j}$ are isomorphic. Thus $\mathcal{F}_{i'}$ is finite locally constant. The proof in cases (2) and (3) is exactly the same. $\square$