The local simplicial model retains actual sections on a finite-dimensional noetherian Nisnevich site

Choose a finite noetherian affine open cover $U_i=\mathrm{Spec}{R}_i$ of $X$. By [F5], every open subset of each $U_i$ is quasi-compact. Every open of $X$ is therefore a finite union of quasi-compact opens obtained by intersecting it with the $U_i$, and is quasi-compact. Thus intersections of affine opens are quasi-compact, so $X$ is qcqs as required by [F1]. Put $d=\dim X$ when $X$ is nonempty and discard empty charts. The prime/closed-subset correspondence in [F5] identifies the ring dimension with the chart dimension. Each $R_i$ consequently has dimension at most $d$: taking closures in $X$ carries a strict chain of irreducible closed subsets of $U_i$ to a chain of the same length, since intersecting those closures with $U_i$ recovers the original chain. The presheaf represented by $U_i$ is a sheaf: factorization through this open is unique, and holds on $T$ exactly when it holds after a covering family, since that family is jointly surjective on points. The slice site over $U_i$ is the small Nisnevich site of $U_i$; its objects and arrows are the same étale schemes with their structure maps, and its covering condition is unchanged. This also identifies the corresponding sheaf-topos slice: a sheaf over the represented $U_i$ is sent on each arrow $T\to U_i$ to its homotopy fiber over that arrow, and the inverse takes the disjoint union of these fibers over the maps $T\to U_i$. Both constructions preserve covering descent. Apply the comparison input of [F1] first from affine objects to absolutely separated objects and then from separated objects to all qcqs étale objects, now with their induced Nisnevich topologies. For the first inclusion, covering arrows between affines and all their finite fiber products over an affine target are affine. Pulling back a covering sieve to an affine object and refining its sources by affine opens gives an induced Nisnevich cover. Each separated object has a finite affine open cover with affine intersections. For the second inclusion, fiber products of separated schemes are separated by [F1]; the same covering-refinement argument applies, and every qcqs object has a finite affine open cover whose intersections are quasi-compact opens of an affine, hence separated. These check all three covering and Čech-product hypotheses of that comparison. Thus this slice is computed by the affine étale $R_i$-site, and hence by ${\mathrm{T}\mathrm{e}\mathrm{s}\mathrm{t}}_{R_i}$ using [F2]. The same identification holds for slices at its represented affine étale test schemes. Theorem 2.24 therefore bounds their homotopy dimensions by $d$.

The represented affine étale test schemes mapping into some $U_i$ generate all sheaves under colimits: cover any test scheme first by the inverse images of the $U_i$ and then by finite affine open refinements from [F1]. For completeness, an intersection in such an affine open cover is a qcqs open of an affine and hence separated by [F1]. It has a finite affine cover with affine intersections, all still mapping into the relevant $U_i$. Its represented sheaf is therefore generated by these affine represented sheaves via the covering Čech colimit. The first cover's Čech colimit then generates every represented test sheaf, and represented test sheaves generate by [F3]. Here a represented sheaf means the ordinary sheafification of the represented presheaf, so this generation argument does not require a separate subcanonicity premise. Step 1.1 bounds the homotopy dimension of every slice at these generators by $d$, so the topos is locally of homotopy dimension at most $d$. Corollary 7.2.1.12 in [F3] makes it hypercomplete. Proposition 6.5.2.14 identifies the local simplicial-presheaf model with this ordinary sheaf category. The smallness and topology in [A1, F1] also meet [F4]. Hence $L^2⊣i$ gives the additional local injective simplicial-sheaf comparison. More explicitly, all objects are cofibrant; for a simplicial presheaf $P$ and a fibrant simplicial sheaf $G$, $iG$ is fibrant and $\mathrm{Map}(L^{2}P,G)=\mathrm{Map}(P,iG)$ as simplicial mapping objects. This enriched adjunction follows by applying degreewise sheafification's adjunction to $P\times {\mathrm{\Delta }}^n$; the sheaf tensor is its associated sheaf. Together with the local-equivalence unit and isomorphism counit of [F4], it identifies derived mapping spaces in the two models. Thus for a simplicial presheaf $P$ representing an ordinary sheaf $F$, a sheaf fibrant replacement $L^{2}P\to G$ has adjoint $P\to iG$ a local equivalence. Under the presheaf comparison both represent $F$. For each test object $T$, write $h_T(R)={\mathrm{Hom}}_X(R,T)$ for its represented presheaf. The simplicial Yoneda identity $\mathrm{Map}(h_T,iG)=(iG)(T)$ computes the derived value, since $h_T$ is cofibrant and $iG$ fibrant; it is the same space as $F(T)$ under the presheaf identification in [F3]. This proves the claimed preservation of actual derived values and mapping classes in both models. Pointed objects and their maps are obtained by taking the category under the terminal sheaf; the equivalence preserves that terminal object and hence preserves the pointed mapping spaces as well. If $X$ is empty, the empty covering forces every sheaf value to be contractible, and the same conclusion holds in the terminal category.