Presheaves and sheaves of sets on a site

Let $X_{\acute{e}t}$ be the small étale site of $X$, with category and covering families fixed in the earlier site definitions. A set-valued presheaf $F$ on this site assigns a set $F(U)$ to every object $U\to X$ and a restriction map $F(U)\to F(V)$ to every arrow $V\to U$, preserving identity maps and composition. For a declared cover $(U_i\to U{)}_{i\in I}$, a matching family is a tuple $s_i\in F(U_i)$ whose two restrictions to every fibre product $U_i{\times }_U{U}_j$ agree. The presheaf is a sheaf if, for each declared cover, every matching family is the restriction of exactly one element of $F(U)$.

The small étale site of a scheme

Jointly surjective étale families form a Grothendieck pretopology

The Stacks Project, Definitions 59.27.1 and 59.27.3

Milne, Lectures on Étale Cohomology, Chapter I, §§5–6

The matching condition uses the actual fibre products in schemes; it does not replace them by intersections of underlying point sets. Assume full AC, as in The Axiom of Choice, when invoking the pretopology certification of Jointly surjective étale families form a Grothendieck pretopology. Under that hypothesis the declared families form a pretopology, so checking the matching-family condition on them is the sheaf condition for the topology they generate: a matching assignment on a covering sieve restricts to a matching family on a generating declared cover, whose unique amalgamation then agrees on every arrow of the sieve by pulling back that cover and using uniqueness. The presheaf and matching-family definitions themselves make no choice of sections.