Lemma 9.1.7.18. Let $\lambda $ and $\mu $ be regular cardinals satisfying $\lambda \trianglelefteq \mu $, and let $X_{\bullet }$ be a $\mu $-small simplicial set. Then there is a $\mu $-small collection $\{ X(i)_{\bullet } \} _{i \in I}$, where each $X(i)_{\bullet }$ is a $\lambda $-small simplicial subset of $X_{\bullet }$, and every $\lambda $-small simplicial subset of $X_{\bullet }$ is contained in some $X(i)_{\bullet }$.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. If $\kappa = \aleph _0$, we can take $\{ X(i)_{\bullet } \} $ to be the collection of all finite simplicial subsets of $X_{\bullet }$. We may therefore assume without loss of generality that $\kappa $ is uncountable. Let $S = \coprod _{n \geq 0} X_{n}$ denote the collection of all simplices of $X$. The assumption $\kappa \trianglelefteq \lambda $ guarantees that there is a $\lambda $-small collection $\{ S(i) \} _{i \in I}$ of $\kappa $-small subsets of $S$ such that every $\kappa $-small subset of $S$ is contained in some $S(i)$. Enlarging the subsets $S(i)$ if necessary, we may assume that they are closed under the face and degeneracy operators for the simplicial set $X_{\bullet }$. It follows that each $S(i)$ can be identified with the collection of simplices for some $\kappa $-small simplicial subset $X(i)_{\bullet } \subseteq X$. By construction, the collection of simplicial subsets $\{ X(i)_{\bullet } \} _{i \in I}$ has the desired property. $\square$