Remark 4.7.4.9. Let $\kappa $ be an infinite cardinal and let $S$ be a $\kappa $-small simplicial set. Suppose we are given a morphism of simplicial sets $f: X \rightarrow S$ and an infinite cardinal $\lambda $ of cofinality $\geq \kappa $. Assume that, for every nondegenerate simplex $\sigma : \Delta ^ n \rightarrow S$, the fiber product $X_{\sigma } = \Delta ^ n \times _{S} X$ is $\lambda $-small. Then $X$ is $\lambda $-small. This follows from Remarks 4.7.4.5 and 4.7.4.8, since $X$ can be realized as a quotient of the disjoint union $\coprod _{\sigma } X_{\sigma }$.
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