Corollary 7.4.3.8 (Size Estimates for Colimits). Let $\lambda $ be an uncountable regular cardinal and let $\kappa = \mathrm{cf}(\lambda )$ be the cofinality of $\kappa $. Then the $\infty $-category $\operatorname{\mathcal{S}}^{< \lambda }$ admits $\kappa $-small colimits, which are preserved by the inclusion functors $\operatorname{\mathcal{S}}^{< \lambda } \hookrightarrow \operatorname{\mathcal{S}}^{< \mu }$ for any $\mu \geq \lambda $.
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Proof. Let $\operatorname{\mathcal{C}}$ be a $\kappa $-small simplicial set; we wish to show that every diagram $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}^{< \lambda }$ admits a colimit (which is preserved by the inclusion functors $\operatorname{\mathcal{S}}^{< \lambda } \hookrightarrow \operatorname{\mathcal{S}}^{< \mu }$ for $\mu \geq \lambda $). Choose a left fibration $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ having $\mathscr {F}$ as its covariant transport representation (for example, we can take $\operatorname{\mathcal{E}}$ to be the $\infty $-category of elements $\int _{\operatorname{\mathcal{C}}} \mathscr {F}$ (see Definition 5.6.2.1). Since $U$ is essentially $\lambda $-small and $\operatorname{\mathcal{C}}$ is $\kappa $-small, the $\infty $-category $\operatorname{\mathcal{E}}$ is essentially $\lambda $-small (Proposition 4.7.9.10). We can therefore choose a weak homotopy equivalence $\operatorname{\mathcal{E}}\rightarrow X$, where $X$ is a $\lambda $-small Kan complex. Applying Proposition 7.4.3.6, we conclude that $X$ is a colimit of $\mathscr {F}$. $\square$