Example 9.1.9.5. If $K$ is a $\kappa $-small simplicial set, then the limit functor $\varprojlim : \operatorname{Fun}(K, \operatorname{\mathcal{S}}) \rightarrow \operatorname{\mathcal{S}}$ is $\kappa $-finitary: this is a refomulation of Theorem 9.1.5.7 (see Proposition 9.1.4.1). In particular, if $K$ is a finite simplicial set, then the functor $\varprojlim : \operatorname{Fun}(K, \operatorname{\mathcal{S}}) \rightarrow \operatorname{\mathcal{S}}$ is finitary.
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