If we let $\mathscr F$ be a sheaf on $X$ let $Et(\mathscr F)$ denote its Étale space over $X$. Is there a way to associate sheaves $\mathscr G$ on $Et(\mathscr F)$ to a sheaf $\mathscr G'$ on $X$ and a natural transformation $\mathscr G' \rightarrow \mathscr F$? So a functor $\text{Sh}(Et(\mathscr F)) \rightarrow \text{Sh}(X)/\mathscr F$? Let me explain what my thoughts are.
If $\mathscr G$ is a sheaf on $Et(\mathscr F)$ we can construct the Étale space of $\mathscr G$ over $Et(\mathscr F)$, $Et(\mathscr G)$. There is an Étale morphism $\phi$ given by composing $Et(\mathscr G) \rightarrow Et(\mathscr F) \rightarrow X$, then let $\mathscr G'$ denote the sheaf of local sections of $\phi$. There is a natural map $Et(\mathscr G') \xrightarrow \approx Et(\mathscr G)$ which gives a map $Et(\mathscr G') \rightarrow Et(\mathscr F)$ over $X$ and so a natural transformation $\mathscr G' \rightarrow \mathscr F$. Does this look correct? And is this a special case of some larger categorical framework?