The Stacks project

Proof. Let us first prove this when $X$ and $X_ i$ are quasi-compact and quasi-separated for all $i$ (as this is true in all cases of interest). In this case any object of $X_{affine, {\acute{e}tale}}$, resp. $(X_ i)_{affine, {\acute{e}tale}}$ is of finite presentation over $X$. Moreover, the category of schemes of finite presentation over $X$ is the colimit of the categories of schemes of finite presentation over $X_ i$, see Limits, Lemma 32.10.1. The same holds for the subcategories of affine objects étale over $X$ by Limits, Lemmas 32.4.13 and 32.8.10. Finally, if $\{ U^ j \to U\}$ is a covering of $X_{affine, {\acute{e}tale}}$ and if $U_ i^ j \to U_ i$ is morphism of affine schemes étale over $X_ i$ whose base change to $X$ is $U^ j \to U$, then we see that the base change of $\{ U^ j_ i \to U_ i\}$ to some $X_{i'}$ is a covering for $i'$ large enough, see Limits, Lemma 32.8.15.

In the general case, let $U$ be an object of $X_{affine, {\acute{e}tale}}$. Then $U \to X$ is étale and separated (as $U$ is separated) but in general not quasi-compact. Still, $U \to X$ is locally of finite presentation and hence by Limits, Lemma 32.10.5 there exists an $i$, a quasi-compact and quasi-separated scheme $U_ i$, and a morphism $U_ i \to X_ i$ which is locally of finite presentation whose base change to $X$ is $U \to X$. Then $U = \mathop{\mathrm{lim}}\nolimits _{i' \geq i} U_{i'}$ where $U_{i'} = U_ i \times _{X_ i} X_{i'}$. After increasing $i$ we may assume $U_ i$ is affine, see Limits, Lemma 32.4.13. To check that $U_ i \to X_ i$ is étale for $i$ sufficiently large, choose a finite affine open covering $U_ i = U_{i, 1} \cup \ldots \cup U_{i, m}$ such that $U_{i, j} \to U_ i \to X_ i$ maps into an affine open $W_{i, j} \subset X_ i$. Then we can apply Limits, Lemma 32.8.10 to see that $U_{i, j} \to W_{i, j}$ is étale after possibly increasing $i$. In this way we see that the functor $\mathop{\mathrm{colim}}\nolimits (X_ i)_{affine, {\acute{e}tale}} \to X_{affine, {\acute{e}tale}}$ is essentially surjective. Fully faithfulness follows directly from the already used Limits, Lemma 32.10.5. The statement on coverings is proved in exactly the same manner as done in the first paragraph of the proof. $\square$