Every presheaf of sets on the small étale site has an associated sheaf

For an object $U$, consider pairs $(\mathcal{U},s)$ where $\mathcal{U}$ is a declared covering family of $U$ and $s$ is a matching family in $F$ on $\mathcal{U}$. Identify two pairs when their restrictions agree on a common covering refinement; pullback stability and composition in [F1] give a common refinement whenever required. The set-sized universe bounds the collections of covers and sections, so the equivalence classes form a set $F^{+}(U)$. Pullback of a pair defines restriction maps, making $F^{+}$ a presheaf.

The natural map $F\to F^{+}$ sends a section to its restrictions on the identity cover. Two sections of $F^{+}(U)$ that agree after restriction to a covering family have representatives which, after pulling back to that family, agree on covering refinements. Composing these refinements with the original cover gives a common covering refinement of $U$, so the two representatives are equivalent. Thus $F^{+}$ is separated: sections of $F^{+}(U)$ agreeing on a cover are equal.

Let $(U_i\to U{)}_{i\in I}$ be a declared cover and let $(s_i{)}_{i\in I}$ be a matching family in $F^{++}$. By [A1], choose for every $i$ a representative of $s_i$ consisting of a declared cover $(V_{i\alpha }\to U_i{)}_{\alpha \in A_i}$ and sections $t_{i\alpha }\in F^{+}(V_{i\alpha })$. The composite family $(V_{i\alpha }\to U{)}_{i,\alpha }$ covers $U$ by transitivity in [F1]. Fix $(i,\alpha )$ and $(j,\beta )$ and put $W=V_{i\alpha }{\times }_U{V}_{j\beta }$. The matching condition for $s_i,s_j$ is an equality in $F^{++}$ on $U_i{\times }_U{U}_j$. By the definition of equality in the plus construction, after pulling the representative covers to this overlap, that equality is witnessed by a common covering refinement on which the corresponding sections of $F^{+}$ agree. By [A1], choose these set-indexed witnesses simultaneously for all pairs $(i,\alpha ),(j,\beta )$. Pulling each witness back to $W$ gives a cover of $W$ on which the restrictions of $t_{i\alpha }$ and $t_{j\beta }$ agree in $F^{+}$. Step 2.1 says $F^{+}$ is separated, so these restrictions were already equal on all of $W$. Thus the family $(t_{i\alpha }{)}_{i,\alpha }$ is matching in $F^{+}$ for the composite cover of $U$, and its representative defines an element of $F^{++}(U)$. For each fixed $i$, the restriction of this representative to $U_i$ is represented by the covering family $(V_{j\beta }{\times }_U{U}_i\to U_i{)}_{j,\beta }$. The family $(V_{i\alpha }{\times }_U{V}_{j\beta }\to U_i{)}_{\alpha ,j,\beta }$ is a common covering refinement of this family and the chosen cover $(V_{i\alpha }\to U_i)$ by pullback stability and composition in [F1]. On each member the pulled-back sections agree by the pairwise equality already proved, so the representatives are equivalent and the restriction is $s_i$. The same plus-separatedness argument of step 2.1, now applied to the presheaf $F^{+}$, says $F^{++}$ is separated, so this amalgamation is unique. Therefore $F^{++}$ is a sheaf.

A presheaf map $F\to G$ into a sheaf sends matching families to matching families and therefore defines a map $F^{+}\to G$ by the unique gluing property in [F2]; agreement on refinements makes this map well-defined. Applying the same construction once more extends it uniquely from $F^{++}$ to $G$. Uniqueness follows because every plus section is locally represented by a matching family from the preceding presheaf and a sheaf map is determined by its restrictions to a cover. This is the stated universal property.