This lemma is a generalization of Groupoids, Lemma \ref{groupoids-lemma-colimit-kappa} which deals with the case of a groupoid in schemes. We can't quite use the same argument, so we use the material on ``crystals of quasi-coherent sheaves'' we developed above. \medskip\noindent Choose a scheme $W$ and a surjective \'etale morphism $W \to U$. Choose a scheme $V$ and a surjective \'etale morphism $V \to W \times_{U, s} R$. Choose a scheme $V'$ and a surjective \'etale morphism $V' \to R \times_{t, U} W$. Consider the collection of schemes $$ I = \{W, W \times_U W, V, V', V \times_R V'\} $$ and the set of morphisms of schemes $$ \Phi = \{\text{pr}_i : W \times_U W \to W, V \to W, V' \to W, V \times_R V' \to V, V \times_R V' \to V'\} $$ Set $X = (I, \Phi)$. Recall that we have defined a category $\textit{CQC}(X)$ of crystals of quasi-coherent sheaves on $X$. There is a functor $$ \QCoh(U, R, s, t, c) \longrightarrow \textit{CQC}(X) $$ which assigns to $(\mathcal{F}, \alpha)$ the sheaf $\mathcal{F}|_W$ on $W$, the sheaf $\mathcal{F}|_{W \times_U W}$ on $W \times_U W$, the pullback of $\mathcal{F}$ via $V \to W \times_{U, s} R \to W \to U$ on $V$, the pullback of $\mathcal{F}$ via $V' \to R \times_{t, U} W \to W \to U$ on $V'$, and finally the pullback of $\mathcal{F}$ via $V \times_R V' \to V \to W \times_{U, s} R \to W \to U$ on $V \times_R V'$. As comparison maps $\{\alpha_\phi\}_{\phi \in \Phi}$ we use the obvious ones (coming from associativity of pullbacks) except for the map $\phi = \text{pr}_{V'} : V \times_R V' \to V'$ we use the pullback of $\alpha : t^*\mathcal{F} \to s^*\mathcal{F}$ to $V \times_R V'$. This makes sense because of the following commutative diagram $$ \xymatrix{ & V \times_R V' \ar[ld] \ar[rd] \\ V \ar[rd] \ar[dd] & & V' \ar[ld] \ar[dd] \\ & R \ar@<-1ex>[dd]_s \ar@<1ex>[dd]^t \\ W \ar[rd] & & W \ar[ld] \\ & U } $$ The functor displayed above isn't an equivalence of categories. However, since $W \to U$ is surjective \'etale it is faithful\footnote{In fact the functor is fully faithful, but we won't need this.}. Since all the morphisms in the diagram above are flat we see that it is an exact functor of abelian categories. Moreover, we claim that given $(\mathcal{F}, \alpha)$ with image $(\{\mathcal{F}_i\}_{i \in I}, \{\alpha_\phi\}_{\phi \in \Phi})$ there is a $1$-to-$1$ correspondence between quasi-coherent submodules of $(\mathcal{F}, \alpha)$ and $(\{\mathcal{F}_i\}_{i \in I}, \{\alpha_\phi\}_{\phi \in \Phi})$. Namely, given a submodule of $(\{\mathcal{F}_i\}_{i \in I}, \{\alpha_\phi\}_{\phi \in \Phi})$ compatibility of the submodule over $W$ with the projection maps $W \times_U W \to W$ will guarantee the submodule comes from a quasi-coherent submodule of $\mathcal{F}$ (by Properties of Spaces, Proposition \ref{spaces-properties-proposition-quasi-coherent}) and compatibility with $\alpha_{\text{pr}_{V'}}$ will insure this subsheaf is compatible with $\alpha$ (details omitted). \medskip\noindent Choose a cardinal $\kappa$ as in Lemma \ref{lemma-crystals-in-quasi-coherent-modules} for the system $X = (I, \Phi)$. It is clear from Properties, Lemma \ref{properties-lemma-set-of-iso-classes} that there is a set of isomorphism classes of $\kappa$-generated crystals in quasi-coherent sheaves on $X$. Hence the result is clear.