The "Galois Correspondence" for n-Stacks

We prove an essentially surjective Galois-correspondence-like functor for $n$-stacks. More specifically, it gives an essentially surjective functor from the $\mathrm{\infty }$-category of $n$-stacks of finite sets with an action of the fundamental group of $X$ to the $\mathrm{\infty }$-category of Deligne-Mumford $n$-stacks finite étale over a connected scheme $X$.