Covering axioms for the common small Nisnevich site

An isomorphism is an open immersion, hence is étale and preserves residue fields by [F2], so its singleton family covers. The identity gives $X\mathrm{/}X$, and any open subscheme of an object is étale over $X$ by [F2]; restricting its sheaf keeps its data in $U$ by [A1]. For a cover $\{T_i\to T{\}}_{i\in I}$ and covers $\{T_{ij}\to T_i{\}}_{j\in J_i}$, the composite arrows are étale by [F2]. Given $t\in T$, first take a residue-field-preserving lift $t_i$, then such a lift $t_{ij}$ of $t_i$. Their composite residue-field map is an isomorphism by [F4], proving the point-lifting condition. The set of pairs $(i,j)$ belongs to $U$ by [A1], so this is a specified covering family. No simultaneous choice of lifts at all points is needed. The empty scheme is an open subscheme, and its empty family covers by the vacuous point condition. The collection of object and arrow encodings is a subset of the set $U$, hence is in $V$ by [A1]; each specified family has an encoding in $U$, so the collection of such families is also $V$-small. For a Zariski open cover of $T$, the distinct open subsets form a subset of the $U$-small topology of $T$ and hence a $U$-small set. For each distinct open choose a representing cover index using [A1]; the corresponding open subobjects of $T$ give a refinement, with a map to that representative. These arrows are étale and preserve residue fields by [F2], so they are a specified covering, including the empty cover of the empty object.

Let $\{T_i\to T{\}}_{i\in I}$ be a specified covering and $S\to T$ any category arrow. By [F3], choose $U$-data products $P_i=T_i{\times }_{T}S$. The projection $P_i\to S$ is étale by [F2], and its composite with $S\to X$ is étale, so $P_i$ belongs to ${\mathcal{C}}_X$; the scheme universal property gives the category pullback. For $s\in S$, put $t$ equal to its image and choose $t_i$ above $t$ with $a:\kappa (t)\to \kappa (t_i)$ an isomorphism. The inverse of $a$ followed by $\kappa (t)\to \kappa (s)$ gives an embedding $\kappa (t_i)\to \kappa (s)$. By [F4] it defines a map $\mathrm{Spec}\kappa (s)\to T_i$ compatible over $T$ with the canonical map $\mathrm{Spec}\kappa (s)\to S$. The universal property [F3] gives a map to $P_i$, with image $p$ over $s$, and [F4] gives an embedding $e:\kappa (p)\hookrightarrow \kappa (s)$. If $b:\kappa (s)\to \kappa (p)$ is induced by the projection, compatibility gives $eb={\mathrm{id}}_{\kappa (s)}$. For any $z\in \kappa (p)$, injectivity of $e$ and $e(b(e(z)))=e(z)$ give $b(e(z))=z$. Thus $b$ and $e$ are inverse isomorphisms, proving the required lift after base change. The index set remains $I\in U$, so this is a specified cover. These pullbacks, the composition and isomorphism assertions of step 1.1 establish the covering axioms. Their generated Grothendieck topology is precisely the Nisnevich topology of [F1], and all size and refinement assertions were proved in step 1.1.