Proposition 9.1.7.3. Let $\lambda $ be an uncountable cardinal, let $\operatorname{\mathcal{C}}$ be a filtered $\infty $-category, and let $\operatorname{\mathcal{C}}_0 \subseteq \operatorname{\mathcal{C}}$ be a $\lambda $-small simplicial subset. Then $\operatorname{\mathcal{C}}_0$ is contained in a $\lambda $-small simplicial subset $\operatorname{\mathcal{C}}' \subseteq \operatorname{\mathcal{C}}$ which is a filtered $\infty $-category.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. Without loss of generality, we may assume that $\lambda $ is regular (Remark 4.7.4.7), and therefore of uncountable cofinality. As in the proof of Proposition 9.1.7.1, we will construct $\operatorname{\mathcal{C}}'$ as the union of an increasing sequence
\[ \operatorname{\mathcal{C}}_0 \subseteq \operatorname{\mathcal{C}}_1 \subseteq \operatorname{\mathcal{C}}_2 \subseteq \operatorname{\mathcal{C}}_3 \subseteq \cdots , \]
of $\lambda $-small simplicial subsets of $\operatorname{\mathcal{C}}$, where every inner horn in $\operatorname{\mathcal{C}}_{n}$ extends to a simplex of $\operatorname{\mathcal{C}}_{n+1}$. To guarantee that $\operatorname{\mathcal{C}}'$ is a filtered $\infty $-category, it will suffice to ensure that the following additional condition is satisfied (see Lemma 9.1.2.12):
For $m,n \geq 0$, every morphism of simplicial sets $f: \operatorname{\partial \Delta }^{m} \rightarrow \operatorname{\mathcal{C}}_{n}$ admits an extension $\overline{f}: (\operatorname{\partial \Delta }^{m})^{\triangleright } \rightarrow \operatorname{\mathcal{C}}_{n+1}$.
This is possible, since the collection $\{ f: \operatorname{\partial \Delta }^{m} \rightarrow \operatorname{\mathcal{C}}_{n} \} _{m \geq 0}$ is $\lambda $-small (Proposition 4.7.4.10) and each of its members admits an extension $\overline{f}: ( \operatorname{\partial \Delta }^{m} )^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$. $\square$