An element of the left hand side is given by the equivalence classe of a pair $(i, \alpha_i)$ where $i \in I$ and $\alpha_i : \mathcal{G} \to \mathcal{F}_i$ is a morphism of $\mathcal{O}_X$-modules, see Categories, Section \ref{categories-section-directed-colimits}. Postcomposing with the coprojection $p_i : \mathcal{F}_i \to \colim_{i' \in I} \mathcal{F}_{i'}$ we get $\alpha = p_i \circ \alpha_i$ in the right hand side. We obtain a map $$ \colim_i \Hom_X(\mathcal{G}, \mathcal{F}_i) \to \Hom_X(\mathcal{G}, \colim_i \mathcal{F}_i) $$ \medskip\noindent Let us show this map is injective. Let $\alpha_i$ be as above such that $\alpha = p_i \circ \alpha_i$ is zero. By the assumption that $\mathcal{G}$ is of finite presentation, for every $x \in X$ we can choose an open neighbourhood $U_x \subset X$ of $x$ and a finite set $s_{x, 1}, \ldots, s_{x, n_x} \in \mathcal{G}(U_x)$ generating $\mathcal{G}|_{U_x}$. These sections map to zero in the stalk $(\colim_{i'} \mathcal{F}_i)_x = \colim_{i'} \mathcal{F}_{i', x}$. Hence for each $x$ we can pick $i(x) \geq i$ such that after replacing $U_x$ by a smaller open we have that $s_{x, 1}, \ldots, s_{x, n_x}$ map to zero in $\mathcal{F}_{i(x)}(U_x)$. Then $X = \bigcup U_x$. By condition (3) we can refine this open covering by a finite open covering $X = \bigcup_{j \in J} U_j$. For $j \in J$ pick $x_j \in X$ with $U_j \subset U_{x_j}$. Set $i' = \max(i(x_j); j \in J)$. Then $\mathcal{G}|_{U_j}$ is generated by the sections $s_{x_j, k}$ which are mapped to zero in $\mathcal{F}_{i(x)}$ and hence in $\mathcal{F}_{i'}$. Hence the composition $\mathcal{G} \to \mathcal{F}_i \to \mathcal{F}_{i'}$ is zero as desired. \medskip\noindent Proof of surjectivity. Let $\alpha$ be an element of the right hand side. For every point $x \in X$ we may choose an open neighbourhood $U \subset X$ and finitely many sections $s_j \in \mathcal{G}(U)$ which generate $\mathcal{G}$ over $U$ and finitely many relations $\sum f_{kj} s_j = 0$, $k = 1, \ldots, n$ with $f_{kj} \in \mathcal{O}_X(U)$ which generate the kernel of $\bigoplus_{j = 1, \ldots, m} \mathcal{O}_U \to \mathcal{G}$. After possibly shrinking $U$ to a smaller open neighbourhood of $x$ we may assume there exists an index $i \in I$ such that the sections $\alpha(s_j)$ all come from sections $s_j' \in \mathcal{F}_i(U)$. After possibly shrinking $U$ to a smaller open neighbourhood of $x$ and increasing $i$ we may assume the relations $\sum f_{kj} s'_j = 0$ hold in $\mathcal{F}_i(U)$. Hence we see that $\alpha|_U$ lifts to a morphism $\mathcal{G}|_U \to \mathcal{F}_i|_U$ for some index $i \in I$. \medskip\noindent By condition (3) and the preceding arguments, we may choose a finite open covering $X = \bigcup_{j = 1, \ldots, m} U_j$ such that (a) $\mathcal{G}|_{U_j}$ is generated by finitely many sections $s_{jk} \in \mathcal{G}(U_j)$, (b) the restriction $\alpha|_{U_j}$ comes from a morphism $\alpha_j : \mathcal{G} \to \mathcal{F}_{i_j}$ for some $i_j \in I$, and (c) the intersections $U_j \cap U_{j'}$ are all quasi-compact. For every pair $(j, j') \in \{1, \ldots, m\}^2$ and any $k$ we can find we can find an index $i \geq \max(i_j, i_{j'})$ such that $$ \varphi_{i_ji}(\alpha_j(s_{jk}|_{U_j \cap U_{j'}})) = \varphi_{i_{j'}i}(\alpha_{j'}(s_{jk}|_{U_j \cap U_{j'}})) $$ see Sheaves, Lemma \ref{sheaves-lemma-directed-colimits-sections} (2). Since there are finitely many of these pairs $(j, j')$ and finitely many $s_{jk}$ we see that we can find a single $i$ which works for all of them. For this index $i$ all of the maps $\varphi_{i_ji} \circ \alpha_j$ agree on the overlaps $U_j \cap U_{j'}$ as the sections $s_{jk}$ generate $\mathcal{G}$ over this overlap. Hence we get a morphism $\mathcal{G} \to \mathcal{F}_i$ as desired.