Corollary 7.4.3.15. Let $\lambda $ be an uncountable cardinal and let $\kappa = \mathrm{cf}(\lambda )$ be the cofinality of $\lambda $. Let $\operatorname{\mathcal{C}}$ be a $\kappa $-small simplicial set and let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{QC}}$ be a diagram. Suppose that, for each object $C \in \operatorname{\mathcal{C}}$, the $\infty $-category $\mathscr {F}(C)$ is essentially $\lambda $-small. Then the colimit $\varinjlim ( \mathscr {F} )$ is essentially $\lambda $-small.
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Proof. For each vertex $C \in \operatorname{\mathcal{C}}$, the $\infty $-category $\mathscr {F}(C)$ is essentially $\lambda $-small, and is therefore essentially $\tau _{C}^{+}$-small for some infinite cardinal $\tau _{C} < \lambda $ (Corollary 4.7.6.17). Since $\lambda $ has cofinality $\kappa $, the supremum $\tau = \mathrm{sup} \{ \tau _ C \} _{C \in \operatorname{\mathcal{C}}}$ satisfies $\tau < \lambda $. Replacing $\lambda $ by the cardinal $\mathrm{sup} \{ \tau ^{+}, \kappa \} $, we are reduced to proving Corollary 7.4.3.15 in the special case where $\lambda $ is regular. In this case, the desired result follows from Variant 7.4.3.14. $\square$