Restriction to the slice commutes with big-site localization

The formulas in [F1] give an adjunction $j^{\ast }⊣j_{\ast }$. Explicitly, a map $j^{\ast }P\to Q$ yields $P(V)\to P(V{\times }_{S}U)\to Q(V{\times }_{S}U\to U)$. In the other direction, evaluate $P\to j_{\ast }Q$ at $V$ and restrict $Q$ along the graph $(\mathrm{id},f):V\to V{\times }_{S}U$ for each $(V,f)$. Naturality and the projection–graph identities make these constructions inverse, coherently; equivalently the forgetful functor on scheme categories has right adjoint $V\mapsto V{\times }_{S}U$, and the functor-category adjunction is the induced one. Hence this is an adjunction of mapping spaces, not just of isomorphism classes.

If $v:P\to P^{\mathrm{\prime }}$ is a local equivalence on $\mathcal{C}$ and $Q$ is any slice sheaf, then $j_{\ast }Q$ is a sheaf by [F1]. The adjunction gives $\mathrm{Map}(j^{\ast }{P}^{\mathrm{\prime }},Q)\simeq \mathrm{Map}(P^{\mathrm{\prime }},j_{\ast }Q)\simeq \mathrm{Map}(P,j_{\ast }Q)\simeq \mathrm{Map}(j^{\ast }P,Q)$. Thus $j^{\ast }v$ is a local equivalence.

Apply this to the localization unit $P\to L_{\mathcal{C}}P$. Its restriction is a local equivalence, and its target is a slice sheaf by [F1]. The universal property in [F2] identifies it with the slice localization unit. The resulting equivalence is canonical and natural in $P$, so it transports natural maps as well as objects. For hyperdescent, pullback preserves hypercovers and the same descent/adjunction argument applies verbatim.