We first prove the second map is an isomorphism. It is injective by  Properties, Lemma \ref{properties-lemma-sections-over-quasi-compact-open}. Since $\mathcal{F}$ is the union of its coherent submodules, see Properties, Lemma \ref{properties-lemma-quasi-coherent-colimit-finite-type} (and Lemma \ref{lemma-coherent-Noetherian}) we may and do assume that $\mathcal{F}$ is coherent to prove surjectivity. Let $\mathcal{F}_n$ denote the quasi-coherent subsheaf of $\mathcal{F}$ consisting of sections annihilated by $\mathcal{I}^n$, see Properties, Lemma \ref{properties-lemma-sections-over-quasi-compact-open}. Since $\mathcal{F}_1 \subset \mathcal{F}_2 \subset \ldots$ we see that $\mathcal{F}_n = \mathcal{F}_{n + 1} = \ldots $ for some $n \geq 0$ by Lemma \ref{lemma-acc-coherent}. Set $\mathcal{H} = \mathcal{F}_n$ for this $n$. By Artin-Rees (Lemma \ref{lemma-Artin-Rees}) there exists an $c \geq 0$ such that $\mathcal{I}^m\mathcal{F} \cap \mathcal{H} \subset \mathcal{I}^{m - c}\mathcal{H}$. Picking $m = n + c$ we get $\mathcal{I}^m\mathcal{F} \cap \mathcal{H} \subset \mathcal{I}^n\mathcal{H} = 0$. Thus if we set $\mathcal{F}' = \mathcal{I}^m\mathcal{F}$ then we see that $\mathcal{F}' \cap \mathcal{F}_n = 0$ and $\mathcal{F}'|_U = \mathcal{F}|_U$. Note in particular that the subsheaf $(\mathcal{F}')_N$ of sections annihilated by $\mathcal{I}^N$ is zero for all $N \geq 0$. Hence by Properties, Lemma \ref{properties-lemma-sections-over-quasi-compact-open} we deduce that the top horizontal arrow in the following commutative diagram is a bijection: $$ \xymatrix{ \colim_n \Hom_{\mathcal{O}_X}( \mathcal{I}^n, \mathcal{F}') \ar[r] \ar[d] & \Gamma(U, \mathcal{F}') \ar[d] \\ \colim_n \Hom_{\mathcal{O}_X}( \mathcal{I}^n, \mathcal{F}) \ar[r] & \Gamma(U, \mathcal{F}) } $$ Since also the right vertical arrow is a bijection we conclude that the bottom horizontal arrow is surjective as desired. \medskip\noindent Next, we prove the first arrow of the lemma is a bijection. By Lemma \ref{lemma-coherent-Noetherian} the sheaf $\mathcal{G}$ is of finite presentation and hence the sheaf $\mathcal{H} = \SheafHom_{\mathcal{O}_X}(\mathcal{G}, \mathcal{F})$ is quasi-coherent, see Schemes, Section \ref{schemes-section-quasi-coherent}. By definition we have $$ \mathcal{H}(U) = \Hom_{\mathcal{O}_U}(\mathcal{G}|_U, \mathcal{F}|_U) $$ Pick a $\psi$ in the right hand side of the first arrow of the lemma, i.e.,  $\psi \in \mathcal{H}(U)$. The result just proved applies to $\mathcal{H}$ and hence there exists an $n \geq 0$ and an $\varphi : \mathcal{I}^n \to \mathcal{H}$ which recovers $\psi$ on restriction to $U$. By Modules, Lemma \ref{modules-lemma-internal-hom} $\varphi$ corresponds to a map $$ \varphi : \mathcal{I}^{\otimes n} \otimes_{\mathcal{O}_X} \mathcal{G} \longrightarrow \mathcal{F}. $$ This is almost what we want except that the source of the arrow is the tensor product of $\mathcal{I}^n$ and $\mathcal{G}$ and not the product. We will show that, at the cost of increasing $n$, the difference is irrelevant. Consider the short exact sequence $$ 0 \to \mathcal{K} \to \mathcal{I}^n \otimes_{\mathcal{O}_X} \mathcal{G} \to \mathcal{I}^n\mathcal{G} \to 0 $$ where $\mathcal{K}$ is defined as the kernel. Note that $\mathcal{I}^n\mathcal{K} = 0$ (proof omitted). By Artin-Rees again we see that $$ \mathcal{K} \cap \mathcal{I}^m(\mathcal{I}^n \otimes_{\mathcal{O}_X} \mathcal{G}) = 0 $$ for some $m$ large enough. In other words we see that $$ \mathcal{I}^m(\mathcal{I}^n \otimes_{\mathcal{O}_X} \mathcal{G}) \longrightarrow \mathcal{I}^{n + m}\mathcal{G} $$ is an isomorphism. Let $\varphi'$ be the restriction of $\varphi$ to this submodule thought of as a map $\mathcal{I}^{m + n}\mathcal{G} \to \mathcal{F}$. Then $\varphi'$ gives an element of the left hand side of the first arrow of the lemma which maps to $\psi$ via the arrow. In other words we have proved surjectivity of the arrow. We omit the proof of injectivity.