From constructible-category hyperdescent to the actual cohomology complex

Let ${\mathcal{D}}_X=D_{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}}^b(X_{\mathrm{e}\mathrm{t}},\mathrm{\Lambda })$. Its unit object ${\underset{‾}{\mathrm{\Lambda }}}_X$ is bounded constructible, and its pullback along every morphism is the corresponding unit. Input [F1] gives the actual equivalence ${\mathcal{D}}_X\simeq {\lim }_{\mathrm{\Delta }}{\mathcal{D}}_{U_{\bullet }}$ for every arc hypercover. Under it the constant unit and all its shifts are the corresponding compatible systems.

Mapping spaces between compatible objects in a limit of infinity-categories are the limits of their mapping spaces. Apply this to the unit and every shift of the unit. The derived endomorphism spectrum of the unit is $R{\mathrm{Hom}}_{X_{\mathrm{e}\mathrm{t}},\mathrm{\Lambda }}({\underset{‾}{\mathrm{\Lambda }}}_X,{\underset{‾}{\mathrm{\Lambda }}}_X)=R{\mathrm{\Gamma }}_{\mathrm{e}\mathrm{t}}(X,{\underset{‾}{\mathrm{\Lambda }}}_X)$, because internal Hom from the free rank-one coefficient sheaf is the identity. These mapping-space comparisons in all shifts give the equivalence of the entire endomorphism complex with the hypercover totalization. The equivalence retains the coefficient action and actual pullbacks. This is hyperdescent of the specified complex, not just isomorphisms of its groups.