Kerodon

Variant 7.4.5.8 (Size Estimates for Limits). Let $\lambda $ be an uncountable cardinal and let $\kappa = \mathrm{ecf}(\lambda )$ be its exponential 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 limit $\varprojlim (\mathscr {F})$ is also essentially $\lambda $-small. This follows from Corollary 7.4.1.13 and Proposition 7.4.5.1.