Homotopy colimit description of stacks

Let $ F$ be say an Artin stack. If $ p: X \to F$ is an atlas for $ F$ , can we express $ F$ , in the $ \infty$ -category $ Shv^{\acute{et}}(k)$ of higher stacks, as a homotopy colimit over the simplicial diagram $ $ … \rightrightarrows X \times_{F} X \rightrightarrows X $ $ in $ Shv ^{\acute{et}}(k)$ ? I ask this because we know that if $ U_{\bullet} \to X$ is a hypercover, for $ X$ a scheme, this fact is built into the very definition of $ Shv ^{\acute{et}}(k)$ . Indeed, by Dugger, Hollander, Isaksen, (in the language of model categories) one obtains $ Shv ^{\acute{et}}(k)$ as a Bousfield localization of $ sPre(k)$ (the category of simplicial presheaves) by inverting morphisms $ $ hocolim_{\Delta} U_{\bullet} \to X $ $

So I’m wondering if we can deduce this fact about a more general higher sheaf or stack $ F$ from the scheme case.