In the statement of the lemma and in this proof a {\it submodule} of a quasi-coherent module $(\mathcal{F}, \alpha)$ is a quasi-coherent submodule $\mathcal{G} \subset \mathcal{F}$ such that $\alpha(t^*\mathcal{G}) = s^*\mathcal{G}$ as subsheaves of $s^*\mathcal{F}$. This makes sense because since $s, t$ are flat the pullbacks $s^*$ and $t^*$ are exact, i.e., preserve subsheaves. The proof will be a repeat of the proof of Properties, Lemma \ref{properties-lemma-colimit-kappa}. We urge the reader to read that proof first. \medskip\noindent Choose an affine open covering $U = \bigcup_{i \in I} U_i$. For each pair $i, j$ choose affine open coverings $$ U_i \cap U_j = \bigcup\nolimits_{k \in I_{ij}} U_{ijk} \quad\text{and}\quad s^{-1}(U_i) \cap t^{-1}(U_j) = \bigcup\nolimits_{k \in J_{ij}} W_{ijk}. $$ Write $U_i = \Spec(A_i)$, $U_{ijk} = \Spec(A_{ijk})$, $W_{ijk} = \Spec(B_{ijk})$. Let $\kappa$ be any infinite cardinal $\geq$ than the cardinality of any of the sets $I$, $I_{ij}$, $J_{ij}$. \medskip\noindent Let $(\mathcal{F}, \alpha)$ be a quasi-coherent module on $(U, R, s, t, c)$. Set $M_i = \mathcal{F}(U_i)$, $M_{ijk} = \mathcal{F}(U_{ijk})$. Note that $$ M_i \otimes_{A_i} A_{ijk} = M_{ijk} = M_j \otimes_{A_j} A_{ijk} $$ and that $\alpha$ gives isomorphisms $$ \alpha|_{W_{ijk}} : M_i \otimes_{A_i, t} B_{ijk} \longrightarrow M_j \otimes_{A_j, s} B_{ijk} $$ see Schemes, Lemma \ref{schemes-lemma-widetilde-pullback}. Using the axiom of choice we choose a map $$ (i, j, k, m) \mapsto S(i, j, k, m) $$ which associates to every $i, j \in I$, $k \in I_{ij}$ or $k \in J_{ij}$ and $m \in M_i$ a finite subset $S(i, j, k, m) \subset M_j$ such that we have $$ m \otimes 1 = \sum\nolimits_{m' \in S(i, j, k, m)} m' \otimes a_{m'} \quad\text{or}\quad \alpha(m \otimes 1) = \sum\nolimits_{m' \in S(i, j, k, m)} m' \otimes b_{m'} $$ in $M_{ijk}$ for some $a_{m'} \in A_{ijk}$ or $b_{m'} \in B_{ijk}$. Moreover, let's agree that $S(i, i, k, m) = \{m\}$ for all $i, j = i, k, m$ when $k \in I_{ij}$. Fix such a collection $S(i, j, k, m)$ \medskip\noindent Given a family $\mathcal{S} = (S_i)_{i \in I}$ of subsets $S_i \subset M_i$ of cardinality at most $\kappa$ we set $\mathcal{S}' = (S'_i)$ where $$ S'_j = \bigcup\nolimits_{(i, j, k, m)\text{ such that }m \in S_i} S(i, j, k, m) $$ Note that $S_i \subset S'_i$. Note that $S'_i$ has cardinality at most $\kappa$ because it is a union over a set of cardinality at most $\kappa$ of finite sets. Set $\mathcal{S}^{(0)} = \mathcal{S}$, $\mathcal{S}^{(1)} = \mathcal{S}'$ and by induction $\mathcal{S}^{(n + 1)} = (\mathcal{S}^{(n)})'$. Then set $\mathcal{S}^{(\infty)} = \bigcup_{n \geq 0} \mathcal{S}^{(n)}$. Writing $\mathcal{S}^{(\infty)} = (S^{(\infty)}_i)$ we see that for any element $m \in S^{(\infty)}_i$ the image of $m$ in $M_{ijk}$ can be written as a finite sum $\sum m' \otimes a_{m'}$ with $m' \in S_j^{(\infty)}$. In this way we see that setting $$ N_i = A_i\text{-submodule of }M_i\text{ generated by }S^{(\infty)}_i $$ we have $$ N_i \otimes_{A_i} A_{ijk} = N_j \otimes_{A_j} A_{ijk} \quad\text{and}\quad \alpha(N_i \otimes_{A_i, t} B_{ijk}) = N_j \otimes_{A_j, s} B_{ijk} $$ as submodules of $M_{ijk}$ or $M_j \otimes_{A_j, s} B_{ijk}$. Thus there exists a quasi-coherent submodule $\mathcal{G} \subset \mathcal{F}$ with $\mathcal{G}(U_i) = N_i$ such that $\alpha(t^*\mathcal{G}) = s^*\mathcal{G}$ as submodules of $s^*\mathcal{F}$. In other words, $(\mathcal{G}, \alpha|_{t^*\mathcal{G}})$ is a submodule of $(\mathcal{F}, \alpha)$. Moreover, by construction $\mathcal{G}$ is $\kappa$-generated. \medskip\noindent Let $\{(\mathcal{G}_t, \alpha_t)\}_{t \in T}$ be the set of $\kappa$-generated quasi-coherent submodules of $(\mathcal{F}, \alpha)$. If $t, t' \in T$ then $\mathcal{G}_t + \mathcal{G}_{t'}$ is also a $\kappa$-generated quasi-coherent submodule as it is the image of the map $\mathcal{G}_t \oplus \mathcal{G}_{t'} \to \mathcal{F}$. Hence the system (ordered by inclusion) is directed. The arguments above show that every section of $\mathcal{F}$ over $U_i$ is in one of the $\mathcal{G}_t$ (because we can start with $\mathcal{S}$ such that the given section is an element of $S_i$). Hence $\colim_t \mathcal{G}_t \to \mathcal{F}$ is both injective and surjective as desired.