Remark 7.4.5.7 (Size Estimates for Colimits). Let $\lambda $ be an uncountable cardinal and let $\kappa = \mathrm{cf}(\lambda )$ be its cofinality. Suppose we are given a diagram $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$, where $\operatorname{\mathcal{C}}$ is a $\kappa $-small simplicial set, and that the Kan complex $\mathscr {F}(C)$ is essentially $\lambda $-small for each $C \in \operatorname{\mathcal{C}}$. Then the colimit $\varinjlim (\mathscr {F})$ is also essentially $\lambda $-small. This follows from Corollary 7.4.3.15 and Proposition 7.4.5.1.

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