In the lemma and in this proof a {\it submodule} of $(\{\mathcal{F}_i\}_{i \in I}, \{\alpha_\phi\}_{\phi \in \Phi})$ means the data of a quasi-coherent submodule $\mathcal{G}_i \subset \mathcal{F}_i$ for all $i$ such that $\alpha_\phi(f_\phi^*\mathcal{G}_i) = \mathcal{G}_{i'}$ as subsheaves of $\mathcal{F}_{i'}$ for all $\phi \in \Phi$. This makes sense because since $f_\phi$ is flat the pullback $f^*_\phi$ is exact, i.e., preserves subsheaves. The proof will be a variant to the proof of Properties, Lemma \ref{properties-lemma-colimit-kappa}. We urge the reader to read that proof first. \medskip\noindent We claim that it suffices to prove the lemma in case all the schemes $X_i$ are affine. To see this let $$ J = \coprod\nolimits_{i \in I} \{U \subset X_i\text{ affine open}\} $$ and let \begin{align*} \Psi = & \coprod\nolimits_{\phi \in \Phi} \{ (U, V) \mid U \subset X_i, V \subset X_{i'}\text{ affine open with } f_\phi(U) \subset V \} \\ & \amalg \coprod\nolimits_{i \in I} \{ (U, U') \mid U, U' \subset X_i\text{ affine open with } U \subset U' \} \end{align*} endowed with the obvious map $\Psi \to J \times J$. Then our $(\mathcal{F}, \alpha)$ induces a crystal in quasi-coherent sheaves $(\{\mathcal{H}_j\}_{j \in J}, \{\beta_\psi\}_{\psi \in \Psi})$ on $Y = (J, \Psi)$ by setting $\mathcal{H}_{(i, U)} = \mathcal{F}_i|_U$ for $(i, U) \in J$ and setting $\beta_\psi$ for $\psi \in \Psi$ equal to the restriction of $\alpha_\phi$ to $U$ if $\psi = (\phi, U, V)$ and equal to $\text{id} : (\mathcal{F}_i|_{U'})|_U \to \mathcal{F}_i|_U$ when $\psi = (i, U, U')$. Moreover, submodules of $(\{\mathcal{H}_j\}_{j \in J}, \{\beta_\psi\}_{\psi \in \Psi})$ correspond $1$-to-$1$ with submodules of $(\{\mathcal{F}_i\}_{i \in I}, \{\alpha_\phi\}_{\phi \in \Phi})$. We omit the proof (hint: use Sheaves, Section \ref{sheaves-section-bases}). Moreover, it is clear that if $\kappa$ works for $Y$, then the same $\kappa$ works for $X$ (by the definition of $\kappa$-generated modules). Hence it suffices to proof the lemma for crystals in quasi-coherent sheaves on $Y$. \medskip\noindent Assume that all the schemes $X_i$ are affine. Let $\kappa$ be an infinite cardinal larger than the cardinality of $I$ or $\Phi$. Let $(\{\mathcal{F}_i\}_{i \in I}, \{\alpha_\phi\}_{\phi \in \Phi})$ be an object of $\textit{CQC}(X)$. For each $i$ write $X_i = \Spec(A_i)$ and $M_i = \Gamma(X_i, \mathcal{F}_i)$. For every $\phi \in \Phi$ with $j(\phi) = (i, i')$ the map $\alpha_\phi$ translates into an $A_{i'}$-module isomorphism $$ \alpha_\phi : M_i \otimes_{A_i} A_{i'} \longrightarrow M_{i'} $$ Using the axiom of choice choose a rule $$ (\phi, m) \longmapsto S(\phi, m') $$ where the source is the collection of pairs $(\phi, m')$ such that $\phi \in \Phi$ with $j(\phi) = (i, i')$ and $m' \in M_{i'}$ and where the output is a finite subset $S(\phi, m') \subset M_i$ so that $$ m' = \alpha_\phi\left(\sum\nolimits_{m \in S(\phi, m')} m \otimes a'_m\right) $$ for some $a'_m \in A_{i'}$. \medskip\noindent Having made these choices we claim that any section of any $\mathcal{F}_i$ over any $X_i$ is in a $\kappa$-generated submodule. To see this suppose that we are given a collection $\mathcal{S} = \{S_i\}_{i \in I}$ of subsets $S_i \subset M_i$ each with cardinality at most $\kappa$. Then we define a new collection $\mathcal{S}' = \{S'_i\}_{i \in I}$ with $$ S'_i = S_i \cup \bigcup\nolimits_{(\phi, m'),\ j(\phi) = (i, i'),\ m' \in S_{i'}} S(\phi, m') $$ Note that each $S'_i$ still has cardinality at most $\kappa$. Set $\mathcal{S}^{(0)} = \mathcal{S}$, $\mathcal{S}^{(1)} = \mathcal{S}'$ and by induction $\mathcal{S}^{(n + 1)} = (\mathcal{S}^{(n)})'$. Then set $S_i^{(\infty)} = \bigcup_{n \geq 0} S_i^{(n)}$ and $\mathcal{S}^{(\infty)} = \{S_i^{(\infty)}\}_{i \in I}$. By construction, for every $\phi \in \Phi$ with $j(\phi) = (i, i')$ and every $m' \in S^{(\infty)}_{i'}$ we can write $m'$ as a finite linear combination of images $\alpha_\phi(m \otimes 1)$ with $m \in S_i^{(\infty)}$. Thus we see that setting $N_i$ equal to the $A_i$-submodule of $M_i$ generated by $S_i^{(\infty)}$ the corresponding quasi-coherent submodules $\widetilde{N_i} \subset \mathcal{F}_i$ form a $\kappa$-generated submodule. This finishes the proof.