Hypercovers and derived hyperdescent on the big cdh slice

First check the hypercover geometry. A slice hypercover over $(X,f)$ becomes an ambient augmented simplicial coproduct of representables by forgetting the component maps to $U$. For any test map $g:V\to X$, its matching sections are exactly the slice matching sections at $(V,f\circ g)$ over $g$, since every component map to $U$ is forced by the augmentation. Every covering sieve over $V$ lifts over $U$ by [F1]. Thus local lifts for slice matching maps give local lifts for the ambient matching maps, and the forgotten object is a hypercover. Conversely restriction of set-valued presheaves along $j$ preserves finite limits and coproducts objectwise and sends $h_W$ to $h_{b(W)}$: a map $V\to W$ over $S$ together with the specified $V\to U$ is exactly a map $V\to W{\times }_{S}U$ over $U$. Local surjectivity is preserved because an ambient covering sieve over a test $V$ lifts to the slice with its given map $V\to U$. Restricting an ambient hypercover over $X$ therefore gives the hypercover with components $b(W_{n,a})$ over $b(X)$, including all simplicial maps. Consequently the descent diagram for $j^{\ast }P$ at a slice hypercover is the diagram for $P$ at its forgotten hypercover, and the descent diagram for $j_{\ast }Q$ at an ambient hypercover is the diagram for $Q$ at this restricted hypercover. This proves both preservation assertions, including augmentation maps and the empty-cover condition.

Fix either site $\mathcal{A}$. By [A1], [F2] and [F3], select a small dense hypercover family $\mathcal{H}$ and a small colimit-generating family $\mathcal{G}$ in $E$. The functors ${\mathrm{Map}}_E(G,-)$ for $G\in \mathcal{G}$ jointly detect equivalences: for a map $v$ on which they are equivalences, mapping out of colimits extends this property to every object of $E$, and testing identities in its source and target produces a two-sided inverse to $v$. The presentable presheaf category has evaluation functors ${\mathrm{ev}}_W$, which are accessible and preserve limits because limits and colimits are pointwise. By [F3], let $F_W$ be the left adjoint of ${\mathrm{ev}}_W$; its adjunction is $\mathrm{Map}(F_{W}G,P)\simeq {\mathrm{Map}}_E(G,P(W))$. For each $H\to h_X$ in $\mathcal{H}$ and $G\in \mathcal{G}$, form the map ${\mathrm{colim}}_{[n]\in {\mathrm{\Delta }}^{\mathrm{o}\mathrm{p}}}{\coprod }_{a\in A_n}{F}_{W_{n,a}}G\to F_{X}G$ induced by the augmentation. The $F_W$ vary covariantly in $W$: they are the adjoints of the natural restriction transformations between the evaluation functors, so all simplicial coherences are retained. These maps form a small set. By the localization theorem in [F3] they have a reflector. Mapping them into $P$ and using the displayed adjunction identifies locality with descent for $\mathcal{H}$, tested by every $G\in \mathcal{G}$. For each such $G$, the coherent space-valued presheaf $V\mapsto {\mathrm{Map}}_E(G,P(V))$ then has descent for every hypercover by [F2] and the model comparison in [F3]. Mapping into limits and the detection property therefore give $E$-valued descent for every hypercover. Conversely hyperdescent implies locality for the chosen set. Its reflector is exactly $L_{\mathcal{A}}^{\mathrm{h}\mathrm{y}\mathrm{p}}$ with the asserted mapping-space universal property. Čech nerves of covering families are hypercovers, so this also enforces the stated empty-cover condition. Applying this construction to each site completes the claim, independently of ordinary sheafification.