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A simplicial presheaf is a presheaf on a site (e.g., the category of topological spaces) taking values in simplicial sets (i.e., a contravariant functor from the site to the category of simplicial sets).
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An explicit isomorphism between the 1st Cech cohomology and the 1st hypercohomology
Some hints in the literature led me to an answer, which I find a bit surprising:
One can take $U'_\bullet=\mathrm{cosk}_0(U_\bullet)$ and the $1$-cocycle $\alpha\in Z^1(U_\bullet,A)\subseteq A(U_1)$ d …