Hypersheaves and bases

Let $X$ be a topological space equipped with a basis. We prove that, for every $\mathrm{\infty }$-category $\mathcal{C}$ with limits, the restriction functor from $\mathcal{C}$-valued hypersheaves on $X$ to basic hypersheaves is an equivalence of $\mathrm{\infty }$-categories.