By lemma 4.9 in [Dugger-Hollander-Isaksen](https://arxiv.org/pdf/math/0205027.pdf), a hypercover is defined as an augmented simplicial object $U\_\bullet\to X$ in the category of simplicial presheaves such that each $U\_n$ is a coproduct of representables and each $U\_n\to (cosk\_{n-1} U\_\bullet)\_n$ is a local epimorphism. 


In the answer of this [post](https://mathoverflow.net/questions/177297/hypercovers-of-sheaves-in-classical-and-quasi-categories), it shows the relation of hypercover for simplicial presheaves with hypercovering in $\infty$-topos. But the definition in the question doesn't require $U\_n$ to be a coproduct of representables. Is the condition that $U\_n$ being a coproduct of representables not essential for the definition of hypercover of simplicial presheaves? 



Definition of hypercover for simplicial presheaves and hypercovering in $\infty$-topos
The condition that $U\_n$ being a coproduct of representables is *not essential* for the definition of hypercover of simplicial presheaves. In fact, in Jardine's *Local Homotopy Theory*, a hypercover is just *defined* to be a local trivial fibration (more general class than DHI's), which one can prove to be equivalent (need quite some work, using Lemma 4.8 of Jardine's book) to DHI's definition in their specific form; also, the hypercover descent in DHI is equivalent to Jardine's descent.


