Let $(C,J)$ be a category with a grothendieck topology. For every object $X \in C$ there's (I hope) a **little site** which is the full subcategory of the slice category $C_{/X}$ whose objects are the morphisms that appear in some $J$-covering family of $X$ and whose morphisms are commuting triangles s.t. all edges appear in some $J$-covering family of their terminal vertex. The topology on this category is the one induced from $J$ (the topology is irrelevant to the question but it might be relevant to the solution).

In other words **we have a (pseudo-)functor $J_{/(-)} : C \to \mathsf{Cat}$ which sends an object to the little $J$-site over it.** Suppose now that **$J_{/(-)}$ is a stack for some other grothedieck topology** $I$ on $C$.

Let $\mathcal{F}$ be a presheaf on $C_{/X}$ s.t. the following conditions hold:

- It is a
sheaf for the $I$-topology.- There exists an
$I$-covering family $\{U_i\}$ of $X$ s.t. for every $U_i$ the pullback $\mathcal{F}_{U_i}$ is a sheaf on the big site $C_{/U_i}$equipped with the $J$-topology.

Question:Is $\mathcal{F}$ a sheaf on $C_{/X}$ for the $J$-topology?

In informal terms: *If $J$-coverings can be glued $I$-locally is $J$-locality an $I$-local property?*

Example:Putting $C=Sch$, $J=Etale$ and $I=Zariski$ we get: If $\mathcal{F}$ is both aZariski sheafandZariski locally an Etale sheafis it anEtale sheaf?