Kerodon

Proof. Without loss of generality, we may assume that $\lambda $ is regular (Remark 4.7.4.7), and therefore of uncountable cofinality. We will construct $\operatorname{\mathcal{C}}'$ as the union of a sequence

consisting of $\lambda $-small simplicial subsets of $\operatorname{\mathcal{C}}$. Fix an integer $n \geq 0$, and assume that $\operatorname{\mathcal{C}}_{n}$ has been constructed. Let $\{ \sigma _{\alpha }: \Lambda ^{m}_{i} \rightarrow \operatorname{\mathcal{C}}_{n} \} _{\alpha \in A}$ be the collection of all inner horns in the simplicial set $\operatorname{\mathcal{C}}_{n}$ (so that $0 < i < m$). Since the simplicial set $\operatorname{\mathcal{C}}_{n}$ is $\lambda $-small, the set $A$ is also $\lambda $-small (Proposition 4.7.4.10). Since $\operatorname{\mathcal{C}}$ is an $\infty $-category, each $\sigma _{\alpha }$ can be extended to a simplex $\overline{\sigma }_{\alpha }: \Delta ^{m} \rightarrow \operatorname{\mathcal{C}}$. We then choose $\operatorname{\mathcal{C}}_{n+1}$ to be any $\lambda $-small simplicial subset of $\operatorname{\mathcal{C}}$ which contains $\operatorname{\mathcal{C}}_{n}$ and each of the simplices $\overline{\sigma }_{\alpha }$. It follows immediately from the construction that $\operatorname{\mathcal{C}}' = \bigcup _{n} \operatorname{\mathcal{C}}_{n}$ is a $\lambda $-small $\infty $-category containing $\operatorname{\mathcal{C}}_0$. $\square$