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 $C$. 



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> Let $\mathcal{F}$ be a presheaf on $C\_{/X}$ s.t. the following conditions hold:
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> 1. It is a **sheaf for the $I$-topology.**
> 2. 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.
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> **Question:** Is $\mathcal{F}$ a sheaf on $C\_{/X}$ for the $J$-topology?
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In informal terms: *If $J$-coverings can be glued $I$-locally is $J$-locality an $I$-local property?*



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> **Example:** Putting $C=Sch$, $J=Etale$ and $I=Zariski$ we get:
>  If $\mathcal{F}$ is both a **Zariski sheaf** and **Zariski locally an Etale sheaf** is it an **Etale sheaf**?
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If $J$-coverings can be glued $I$-locally is $J$-locality an $I$-local property? (Reducing descent problems to simpler ones)
The answer is yes. In fact the assumption that $X\mapsto J\_{/X}$ is a sheaf is not even relevant. First, let me point out that strictly speaking a Grothendieck topology $J$ does not determine a "small site" over an object $X$, because every morphism belongs to some $J$-covering sieve (the maximal sieve on any object is always $J$-covering). For example, the étale topology on the category of schemes does not determine the small étale site of a scheme, at least not in any obvious way. The reason small sites are so common is that the corresponding topologies come with [bases](https://ncatlab.org/nlab/show/Grothendieck+pretopology) which determine small sites, e.g., jointly surjective families of étale maps form a basis for the étale topology etc.


Now let me explain why the result holds, assuming only 1 and 2; in particular the "small site" will not matter.
Let $U$ be the $I$-covering sieve on $X$ generated by $\{U\_i\to X\}$. I will view sieves as subpresheaves of representable presheaves, like so: $U \hookrightarrow X$. Let $V$ be a $J$-covering sieve on $Y\in C\_{/X}$. Since $I$-local isomorphisms are stable under pullbacks (sheafification is left exact), $V\times\_XU\hookrightarrow V$ is an $I$-local isomorphism, and similarly $V\times\_XU \hookrightarrow Y\times\_XU$ is a $J$-local isomorphism. Since $F$ is an $I$ sheaf on $C$ by 1 and a $J$-sheaf on $C\_{/U}$ by 2, it sends all three morphisms $Y\times\_XU\hookrightarrow Y$, $V\times\_XU\hookrightarrow V$, $V\times\_XU \hookrightarrow Y\times\_XU$ to isomorphisms. By 2-out-of-3, $F$ also sends $V\hookrightarrow Y$ to an isomorphism. This shows that $F$ is a $J$-sheaf.


