Sheafification is left adjoint to the inclusion of sheaves into presheaves

Let $\sigma \in {\mathcal{F}}^{+}(U)$ be represented by sections $s_i\in \mathcal{F}(U_i)$ on an open cover $U={\bigcup }_i{U}_i$. The sections ${\phi }_{U_i}(s_i)\in \mathcal{G}(U_i)$ have equal germs on overlaps because the $s_i$ do. Since $\mathcal{G}$ is a sheaf, they glue uniquely to a section ${\phi }_U^{+}(\sigma )\in \mathcal{G}(U)$. This is independent of the chosen presentation because equivalent presentations have the same germs at every point, and a sheaf is separated. Thus $\phi$ extends uniquely to a morphism ${\phi }^{+}:{\mathcal{F}}^{+}\to \mathcal{G}$.

Apply the same construction again to the morphism ${\phi }^{+}:{\mathcal{F}}^{+}\to \mathcal{G}$. Because $\mathcal{G}$ is already a sheaf, this yields a morphism $\bar{\phi }:({\mathcal{F}}^{+}{)}^{+}=a\mathcal{F}⟶\mathcal{G}$ whose restriction along the second unit map is ${\phi }^{+}$. Composing with the first unit map gives $\bar{\phi }\circ {\eta }_{\mathcal{F}}=\phi$.

To prove uniqueness, let $\psi ,{\psi }^{\mathrm{\prime }}:a\mathcal{F}\to \mathcal{G}$ satisfy $\psi \circ {\eta }_{\mathcal{F}}={\psi }^{\mathrm{\prime }}\circ {\eta }_{\mathcal{F}}$. Fix an open set $U$ and a section $\sigma \in a\mathcal{F}(U)$. By [F1], $\sigma$ is represented locally by sections of ${\mathcal{F}}^{+}$, and each such local ${\mathcal{F}}^{+}$-section is itself locally represented by sections of $\mathcal{F}$. Therefore $U$ admits an open cover $\{U_i\}$ and sections $s_i\in \mathcal{F}(U_i)$ such that $\sigma {\mathrm{\mid }}_{U_i}={\eta }_{\mathcal{F},U_i}(s_i)$ for every $i$. On each $U_i$ the hypothesis gives ${\psi }_U(\sigma ){\mathrm{\mid }}_{U_i}={\psi }_{U_i}({\eta }_{\mathcal{F},U_i}(s_i))={\psi }_{U_i}^{\mathrm{\prime }}({\eta }_{\mathcal{F},U_i}(s_i))={\psi }_U^{\mathrm{\prime }}(\sigma ){\mathrm{\mid }}_{U_i}\mathrm{.}$ Since $\mathcal{G}$ is a sheaf, locality forces ${\psi }_U(\sigma )={\psi }_U^{\mathrm{\prime }}(\sigma )$. Thus $\psi ={\psi }^{\mathrm{\prime }}$.

Steps 2.1 and 3.1 prove that every morphism $\mathcal{F}\to \mathcal{G}$ factors uniquely through ${\eta }_{\mathcal{F}}$. This is the stated adjoint universal property.