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:



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



Approximation of constructible abelian étale sheaves
<https://stacks.math.columbia.edu/tag/09YU>


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


