Pullback and composition of the cdh generating families

If $y\in Y$ over $x\in X$ has $\kappa (y)=\kappa (x)$, then over any $x^{\mathrm{\prime }}\in X^{\mathrm{\prime }}$ above $x$, the fiber of $Y{\times }_X{X}^{\mathrm{\prime }}$ contains the point specified by the $\kappa (x^{\mathrm{\prime }})$-valued base change of $y$. Its residue field is exactly $\kappa (x^{\mathrm{\prime }})$. Thus complete decomposition survives every base change. Combining this with [F1] proves base-change stability of both sorts of generating families. Identity maps plainly have the lifting property.

To lift a point through a composite family, lift it through the first family with unchanged residue field, then through the covering family above that chosen object, again with unchanged residue field. Composing the two residue-field isomorphisms proves complete decomposition of the composite. A pullback of such a composite is the composite of its pulled-back families by the fiber-product associativity isomorphism. This proves the asserted finite-composition and pullback closure.

A point of a finite disjoint union lies in exactly one component. For covers over disjoint target components apply the required lifting property on that component. For a Nisnevich family over a common target apply its lifting property to one summand of its source coproduct. Étaleness and properness are supplied by [F1]; for the proper assertion the targets are disjoint, so properness is checked on those open-and-closed pieces. This proves the stated coproduct assertions.