Representable presheaves are sheaves on the small étale site

Precomposition preserves identities and composition, so $h_Y$ is a presheaf by the definition in [F1]. Let $(U_i\to U)$ be a declared cover and let $f_i:U_i\to Y$ be a matching family; thus their restrictions agree on every $U_i{\times }_U{U}_j$. It remains to construct and prove uniqueness of a descended $X$-morphism.

Fix $u\in U$. By joint surjectivity choose $i$ and $u_i\in U_i$ over $u$. Let $q:Y\to X$ be the given structure map. Choose an affine open $X_0\subseteq X$ containing the image of $u$, an affine open $V_0\subseteq U$ containing $u$ and mapping into $X_0$, and an affine open $Z\subseteq q^{-1}(X_0)$ containing $f_i(u_i)$. Choose an affine open $W\subseteq U_i$ containing $u_i$ with $f_i(W)\subseteq Z$. Since $U_i\to U$ is étale and hence open by [F2], its image of $W$ is an open neighbourhood of $u$; choose an affine open $V\subseteq U$ containing $u$ and contained in that image and in $V_0$. The affine opens of $W{\times }_{U}V$ cover it, and their images are open by [F2] and cover $V$. Since $V$ is quasi-compact, finitely many such affine opens $T_1,\dots ,T_n$ have images covering $V$. Their finite disjoint union $T$ is affine, and $T\to V$ is affine, flat, and surjective: it is a finite disjoint union of restrictions of étale morphisms. It is therefore faithfully flat. Every map $T_j\to Y$ lands in $Z$, and the maps agree on $T_j{\times }_V{T}_k$ by the matching condition on the original family.

Write $V=\mathrm{Spec}A$, $T=\mathrm{Spec}B$, $X_0=\mathrm{Spec}{A}_0$, and $Z=\mathrm{Spec}C$. The faithfully flat ring map is $A\to B$, and $C$ is an $A_0$-algebra because $Z\subseteq q^{-1}(X_0)$. The compatible maps $T\to Z$ give an $A_0$-algebra map $C\to B$ whose two composites to $B{\otimes }_{A}B$ agree. The canonical faithfully-flat equalizer is $A=\mathrm{Eq}(B\rightrightarrows B{\otimes }_{A}B)$, so every element of $C$ maps into $A$. Thus the map uniquely descends to an $A_0$-algebra map $C\to A$, hence to a unique $X_0$-morphism $V\to Z$ by [F3]. The descended map is over $X_0$, hence is an $X$-morphism $V\to Y$. Its pullback to each $T_j$ is the given $f_i{\mathrm{\mid }}_{T_j}$.

Carry out step 1.2 at every point of $U$. On overlaps, the resulting local maps have the same value at each point: choose a point of an original covering object above it, and use the matching condition on the fibre product together with [F1]. After shrinking around that point, both maps land in one affine open of $Y$. Their pullbacks to a jointly-surjective étale refinement agree there by the original matching condition; affine faithful-flat descent in [F3] gives equality of the local maps. The local morphisms therefore glue on the open cover of $U$ to a morphism $f:U\to Y$. The same local uniqueness proves that no other morphism has the required restrictions.

If $U$ is empty, its unique morphism to $Y$ is the required amalgamation. Otherwise the descended morphism is unique and exists, so every matching family for every declared cover has exactly one amalgamation. By [F1], this is precisely the sheaf condition in [F1]'s pretopology, hence $h_Y$ is a sheaf on $X_{\acute{e}t}$. The argument treats arbitrary schemes $Y$ by reducing to affine target neighbourhoods; it does not assert that an arbitrary sheaf is representable.