Finite coproducts and refinement inside a sieve

Realize the construction of the $B$-family by a finite tree of elementary generating-family substitutions. Start with the frontier $X$. When a leaf $A$ is replaced by a generating family over it, take its source coproduct and leave all other leaves unchanged using their identities. The new frontier coproduct maps to the old one. For a Nisnevich substitution this is a Nisnevich covering morphism; for a cdp substitution it is cdp, by [F1] and the disjoint target components. Finitely many substitutions give a finite word with final frontier $W$, and composite $u$. Hence $u$ is cdh covering. This is a consequence of the family construction, not closure of the sieve $S$ under coproducts.

Under the stated refinement, preimages of the summands of $W$ give a finite open-and-closed decomposition $V={\coprod }_i{V}_i$. Each $V_i\to X^{\mathrm{\prime }}$ is étale since $q$ is. Every point of $X^{\mathrm{\prime }}$ has a residue-preserving lift in $V$ by the Nisnevich condition on $q$, and that point belongs to a summand $V_i$. Thus these restricted maps form a finite Nisnevich family. Their composites to $X$ factor as $V_i\to U_i\stackrel{u_i}{\to }X$ and therefore belong to $S$ by sieve closure under precomposition. The schemes $V_i$ remain separated of finite type over $k$.

For emphasis, the generally invalid implication $u_i\in S$ for all $i\Rightarrow \coprod u_i\in S$ has not been used. For example the sieve on a disconnected $X=A⨿B$ generated by its two nonempty summand inclusions does not contain ${\mathrm{i}\mathrm{d}}_X$, although their coproduct is ${\mathrm{i}\mathrm{d}}_X$. It is nonetheless a Nisnevich covering sieve. The split family in step 1.2 is the required correct refinement inside $S$.