The image sheaf is the sheafification of the presheaf image

The assignment $U\mapsto I(U)$ is a subpresheaf of $\mathcal{G}$: if $t={\phi }_U(s)\in I(U)$ and $V\subseteq U$, then $t{\mathrm{\mid }}_V={\phi }_V(s{\mathrm{\mid }}_V)\in I(V)$ by [F2].

The assignment $J$ is a subsheaf of $\mathcal{G}$. Indeed, restriction clearly preserves local representability. If sections $t_i\in J(U_i)$ on an open cover of $U$ agree on overlaps, then because $\mathcal{G}$ is a sheaf they glue to a unique $t\in \mathcal{G}(U)$. The local witnesses for each $t_i$ also witness that $t$ lies in $J(U)$. Thus $J$ is a sheaf, and [F1] makes it a subsheaf. The factorization $\mathcal{F}\to J$ is immediate from the definition of $J$.

Since $I$ maps into the sheaf $J$, [L1] gives a unique morphism $\alpha :aI⟶J$ extending the inclusion $I\hookrightarrow J$.

Conversely, let $t\in J(U)$. Choose an open cover $U={\bigcup }_i{U}_i$ and sections $s_i\in \mathcal{F}(U_i)$ with ${\phi }_{U_i}(s_i)=t{\mathrm{\mid }}_{U_i}$. Then the sections ${\phi }_{U_i}(s_i)\in I(U_i)$ are compatible on overlaps because they all come from $t$. Hence they define a section ${\beta }_U(t)\in aI(U)$ by the very construction of sheafification. These assignments are compatible with restriction, so they define a morphism $\beta :J\to aI$.

The composites $\alpha \circ \beta$ and $\beta \circ \alpha$ are identities, because both are identities locally on the chosen image sections that generate $J$ and $aI$, and both targets are sheaves. Therefore $\alpha$ is an isomorphism.