In certain concrete situations, I can show that the small étale topos of an inverse limit of schemes is the inverse limit of the associated toposes, for example, if $X$ is a (qcqs) scheme relative over some affine scheme $\operatorname{Spec}(R)$, and $S$ is a filtered colimit of $R$-algebras where $S=\varinjlim S_i$, I can show that there is an equivalence of small étale toposes

$$\operatorname{\acute{E}t}(X\times_{\operatorname{Spec}(R)} \operatorname{Spec}(S))\simeq \varprojlim \operatorname{\acute{E}t}(X\times_{\operatorname{Spec}(R)} \operatorname{Spec}(S_i)).$$

(**Note**: The inverse limit here is being taken in the category of toposes, not in the category of locally ringed toposes!!)

I heard from a friend that there is a general statement that given an inverse system $\{X_i\}$ of schemes with affine transition maps and limit $X$, then we have an equivalence of étale toposes:

$$\operatorname{\acute{E}t}(X)\simeq \varprojlim\operatorname{\acute{E}t}(X_i).$$

I looked high and low for a reference, and I couldn't find a proof of this statement. Can someone provide a reference?

1more comment