The second plus construction is a sheaf

Let $U={\bigcup }_{i\in I}{U}_i$ be an open cover and let ${\sigma }_i\in {\mathcal{F}}^{++}(U_i)$ be compatible on overlaps. For each $i$, choose a presentation of ${\sigma }_i$ by sections ${\tau }_{i,a}\in {\mathcal{F}}^{+}(V_{i,a})$ on an open cover $U_i={\bigcup }_a{V}_{i,a}$.

The family of all pairs $(V_{i,a},{\tau }_{i,a})$ covers $U$. Presentations belonging to one fixed $i$ are germ-compatible by definition. If $x\in V_{i,a}\cap V_{j,c}$, compatibility of ${\sigma }_i$ and ${\sigma }_j$ says that their restrictions to $U_i\cap U_j$ are equal; the definition of equality of plus presentations therefore gives $({\tau }_{i,a}{)}_x=({\tau }_{j,c}{)}_x\text{in }({\mathcal{F}}^{+}{)}_x\mathrm{.}$ Thus the combined family is a germ-compatible presentation by ${\mathcal{F}}^{+}$-sections and defines $\sigma \in ({\mathcal{F}}^{+}{)}^{+}(U)={\mathcal{F}}^{++}(U)$. On each $U_i$, its restricted presentation is equivalent to the chosen presentation of ${\sigma }_i$, so $\sigma {\mathrm{\mid }}_{U_i}={\sigma }_i$.

Apply [L1] to the presheaf ${\mathcal{F}}^{+}$. It says that $({\mathcal{F}}^{+}{)}^{+}={\mathcal{F}}^{++}$ is separated. Hence sections of ${\mathcal{F}}^{++}(U)$ whose restrictions agree on the cover $\{U_i\}$ are equal, so locality and uniqueness of the gluing from step 2.1 hold.

Steps 2.1 and 3.1 give gluing and locality for every open cover. Therefore [F2] shows that ${\mathcal{F}}^{++}$ is a sheaf.