Let $X$ be a topological space with sheaf of abelian groups $\mathcal{F}$. Embed $\mathcal{F}$ into a flasque sheaf $\mathscr{I}$ and take the quotient so we have a short exact sequence,
$$
0 \longrightarrow \mathcal{F} \longrightarrow \mathscr{I} \longrightarrow \mathcal{Q} \longrightarrow 0
$$
For any open subset $V \subseteq X$ with inclusion $j \hookrightarrow X$, let $_V \mathcal{F}$ denote the sheaf $j_{*} (\mathcal{F}|_{V})$. Since $j_{*}$ is left exact, we have an exact sequence,
$$
0 \longrightarrow _V \hspace{-0.1cm} \mathcal{F} \longrightarrow _V \hspace{-0.1cm} \mathscr{I} \longrightarrow _V \hspace{-0.1cm} \mathcal{Q}
$$
We also have a canonical morphism $\mathcal{F} \longrightarrow _V \hspace{-0.1cm} \mathcal{F}$ given by the unit of the adjunction $j^{-1} \dashv j_{*}$ so we get a morphism of chain complexes so that we have the following commutative diagram with exact rows.
My question is about a claim I read that says that the image of $\mathcal{Q}$ in $_V \mathcal{Q}$ and the image of $_V \mathscr{I}$ in $_V \mathcal{Q}$ coincide. Is anyone able to tell me why this is true? I have thought about it for a while and I don't see why it's supposed to be clear. I think it has something to do with the fact that $\mathscr{I}$ is flasque and so $\alpha$ is surjective. I also know that the kernel of $\alpha$ and $\beta$ is the subsheaf of sections supported in the complement of $V$, but I haven't been able to use this fact either.
For reference, this comes from the proof of Proposition 1 in this paper of Kempf.
By assumption $\mathscr{I}$ is flasque, so as you say $\alpha$ is surjective; hence the image of ${\hspace{0pt}}_{V}\mathscr{I} \to \mathcal{Q}$ is the same as the image of the composite $\mathscr{I} \to {\hspace{0pt}}_{V}\mathscr{I} \to \mathcal{Q}$. The map $\mathscr{I} \to \mathcal{Q}$ is surjective by assumption so the image of $\mathcal{Q} \to {\hspace{0pt}}_{V}\mathcal{Q}$ is the same as the image of the composite $\mathscr{I} \to \mathcal{Q} \to {\hspace{0pt}}_{V}\mathcal{Q}$.