Choose a set $I$ and for each $i \in I$ an $\mathcal{O}_X$-module of finite presentation and a homomorphism of $\mathcal{O}_X$-modules $\varphi_i : \mathcal{F}_i \to \mathcal{F}$ with the following property: For any $\psi : \mathcal{G} \to \mathcal{F}$ with $\mathcal{G}$ of finite presentation there is an $i \in I$ such that there exists an isomorphism $\alpha : \mathcal{F}_i \to \mathcal{G}$ with $\varphi_i = \psi \circ \alpha$. It is clear from Modules, Lemma \ref{modules-lemma-set-isomorphism-classes-finite-type-modules} that such a set exists (see also its proof). We denote $\mathcal{I}$ the category with $\Ob(\mathcal{I}) = I$ and given $i, i' \in I$ we set $$ \Mor_\mathcal{I}(i, i') = \{\alpha : \mathcal{F}_i \to \mathcal{F}_{i'} \mid \alpha \circ \varphi_{i'} = \varphi_i \}. $$ We claim that $\mathcal{I}$ is a filtered category and that $\mathcal{F} = \colim_i \mathcal{F}_i$. \medskip\noindent Let $i, i' \in I$. Then we can consider the morphism $$ \mathcal{F}_i \oplus \mathcal{F}_{i'} \longrightarrow \mathcal{F} $$ which is the direct sum of $\varphi_i$ and $\varphi_{i'}$. Since a direct sum of finitely presented $\mathcal{O}_X$-modules is finitely presented we see that there exists some $i'' \in I$ such that $\varphi_{i''} : \mathcal{F}_{i''} \to \mathcal{F}$ is isomorphic to the displayed arrow towards $\mathcal{F}$ above. Since there are commutative diagrams $$ \xymatrix{ \mathcal{F}_i \ar[r] \ar[d] & \mathcal{F} \ar@{=}[d] \\ \mathcal{F}_i \oplus \mathcal{F}_{i'} \ar[r] & \mathcal{F} } \quad \text{and} \quad \xymatrix{ \mathcal{F}_{i'} \ar[r] \ar[d] & \mathcal{F} \ar@{=}[d] \\ \mathcal{F}_i \oplus \mathcal{F}_{i'} \ar[r] & \mathcal{F} } $$ we see that there are morphisms $i \to i''$ and $i' \to i''$ in $\mathcal{I}$. Next, suppose that we have $i, i' \in I$ and morphisms $\alpha, \beta : i \to i'$ (corresponding to $\mathcal{O}_X$-module maps $\alpha, \beta : \mathcal{F}_i \to \mathcal{F}_{i'}$). In this case consider the coequalizer $$ \mathcal{G} = \Coker( \mathcal{F}_i \xrightarrow{\alpha - \beta} \mathcal{F}_{i'} ) $$ Note that $\mathcal{G}$ is an $\mathcal{O}_X$-module of finite presentation. Since by definition of morphisms in the category $\mathcal{I}$ we have $\varphi_{i'} \circ \alpha = \varphi_{i'} \circ \beta$ we see that we get an induced map $\psi : \mathcal{G} \to \mathcal{F}$. Hence again the pair $(\mathcal{G}, \psi)$ is isomorphic to the pair $(\mathcal{F}_{i''}, \varphi_{i''})$ for some $i''$. Hence we see that there exists a morphism $i' \to i''$ in $\mathcal{I}$ which equalizes $\alpha$ and $\beta$. Thus we have shown that the category $\mathcal{I}$ is filtered. \medskip\noindent We still have to show that the colimit of the diagram is $\mathcal{F}$. By definition of the colimit, and by our definition of the category $\mathcal{I}$ there is a canonical map $$ \varphi : \colim_i \mathcal{F}_i \longrightarrow \mathcal{F}. $$ Pick $x \in X$. Let us show that $\varphi_x$ is an isomorphism. Recall that $$ (\colim_i \mathcal{F}_i)_x = \colim_i \mathcal{F}_{i, x}, $$ see Sheaves, Section \ref{sheaves-section-limits-sheaves}. First we show that the map $\varphi_x$ is injective. Suppose that $s \in \mathcal{F}_{i, x}$ is an element such that $s$ maps to zero in $\mathcal{F}_x$. Then there exists a quasi-compact open $U$ such that $s$ comes from $s \in \mathcal{F}_i(U)$ and such that $\varphi_i(s) = 0$ in $\mathcal{F}(U)$. By Lemma \ref{lemma-extend} we can find a finite type quasi-coherent subsheaf $\mathcal{K} \subset \Ker(\varphi_i)$ which restricts to the quasi-coherent $\mathcal{O}_U$-submodule of $\mathcal{F}_i$ generated by $s$: $\mathcal{K}|_U = \mathcal{O}_U\cdot s \subset \mathcal{F}_i|_U$. Clearly, $\mathcal{F}_i/\mathcal{K}$ is of finite presentation and the map $\varphi_i$ factors through the quotient map $\mathcal{F}_i \to \mathcal{F}_i/\mathcal{K}$. Hence we can find an $i' \in I$ and a morphism $\alpha : \mathcal{F}_i \to \mathcal{F}_{i'}$ in $\mathcal{I}$ which can be identified with the quotient map $\mathcal{F}_i \to \mathcal{F}_i/\mathcal{K}$. Then it follows that the section $s$ maps to zero in $\mathcal{F}_{i'}(U)$ and in particular in $(\colim_i \mathcal{F}_i)_x = \colim_i \mathcal{F}_{i, x}$. The injectivity follows. Finally, we show that the map $\varphi_x$ is surjective. Pick $s \in \mathcal{F}_x$. Choose a quasi-compact open neighbourhood $U \subset X$ of $x$ such that $s$ corresponds to a section $s \in \mathcal{F}(U)$. Consider the map $s : \mathcal{O}_U \to \mathcal{F}$ (multiplication by $s$). By Lemma \ref{lemma-extend-finite-presentation} there exists an $\mathcal{O}_X$-module $\mathcal{G}$ of finite presentation and an $\mathcal{O}_X$-module map $\mathcal{G} \to \mathcal{F}$ such that $\mathcal{G}|_U \to \mathcal{F}|_U$ is identified with $s : \mathcal{O}_U \to \mathcal{F}$. Again by definition of $\mathcal{I}$ there exists an $i \in I$ such that $\mathcal{G} \to \mathcal{F}$ is isomorphic to $\varphi_i : \mathcal{F}_i \to \mathcal{F}$. Clearly there exists a section $s' \in \mathcal{F}_i(U)$ mapping to $s \in \mathcal{F}(U)$. This proves surjectivity and the proof of the lemma is complete.