I am interested in relating the definition of hypercovers in the $\infty$-topos of sheaves on an $\infty$-Grothendieck site to the classical definition of hypercovers of presheaves on a Grothendieck site.

The definition for hypercovers in an $\infty$-topos that I am using is from *Higher Topos Theory*: In an $\infty$-topos $\mathfrak{X}$ a hypercovering of an element $X$ is the structure map $|U_\bullet|\to X$ of the geometric realization of a simplicial object $U_\bullet\in s\mathfrak{X}_{/X}$ such that the map $$U_n\to (cosk_{n-1}U_\bullet)_n$$ is an effective epimorphism in $\mathfrak{X}_{/X}$ (its Cech nerve is a simplicial resolution of the target) for all $n\geq 1.$

My question now is if this definition gets any easier if we restrict to an $\infty$-topos that is given by the $\infty$-sheaves on a (small) quasi-category with Grothendieck topology. Ideally, I want to relate this to the classical definition that a hypercovering on the presheaves $\mathcal{P(C)}$ of a Grothendieck site is an augmented simplicial object $U_\bullet\in s_+\mathcal{P(C)}$ (with $X\cong U_{-1}$) such that the maps $$U_n\to (cosk_{n-1}U_\bullet)_n$$ are local epimorphisms.

So it all boils down to the question if there is some way of viewing the effective epimorphisms of the $\infty$-sheaf topos on an $\infty$-site as local epimorphisms. Is there any source that does this example? Lastly, I just want to note that we get $\infty$-sheaves from $\infty$-presheaves by localizing along those local epimorphisms the target of which is representable. But I don't see how this would imply my question.