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Let $F$ be a constructible abelian étale sheaf of modules over a finite ring $\Lambda$ on a scheme $X$ over a field $k$, with the size of $\Lambda$ invertible on $X$.

Suppose $X = \varprojlim X_j$, for $X_j$ qcqs schemes over $k$, with affine transition maps.

Do there exist, for $j$ large enough onwards, $F_j$ constructible abelian étale sheaf of $\Lambda$-modules on $X_j$, such that, calling $p_j : X\to X_j$ the natural map, $p_j^*F_j = F$?

Do morphisms of constructible abelian étale sheaves also descend?

In other words, do we have an equivalence, denoting by $\text{Cons}(X_{et},\Lambda)$ the category of constructible abelian étale sheaves of $\Lambda$-modules:

$2\mbox{-}\text{colim}_j\ \text{Cons}(X_{j, et},\Lambda) = \text{Cons}(X_{et},\Lambda)$?

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    $\begingroup$ I deleted my answer because it was wrong and because you found the correct answer on your own. I suggest you pose the stacks project link and accept it. $\endgroup$
    – Will Sawin
    Commented May 10, 2018 at 12:27

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https://stacks.math.columbia.edu/tag/09YU

I found an answer to my question in part (3) of the Lemma in the SP.

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