Lemma 21.43.6. In the situation above, there exists a cardinal $\kappa $ with the following property: given a complex $\mathcal{F}^\bullet $ of $\mathcal{O}$-modules and subsets $\Omega ^ i_ U \subset \mathcal{F}^ i(U)$ there exists a subcomplex $\mathcal{H}^\bullet \subset \mathcal{F}^\bullet $ with $\Omega ^ i_ U \subset \mathcal{H}^ i(U)$ and $|\mathcal{H}^\bullet | \leq \max (\kappa , |\bigcup \Omega ^ i_ U|)$.
Proof. Define $\mathcal{H}^ i(U)$ to be the $\mathcal{O}(U)$-submodule of $\mathcal{F}^ i(U)$ generated by the images of $\Omega ^ i_ V$ and $\text{d}(\Omega ^{i - 1}_ U)$ by restriction along any morphism $f : U \to V$. The cardinality of $\mathcal{H}^ i(U)$ is bounded by the maximum of $\aleph _0$, the cardinality of the $\mathcal{O}(U)$, the cardinality of $\text{Arrows}(\mathcal{C})$, and $|\bigcup \Omega ^ i_ U|$. Details omitted. $\square$
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