Kerodon

Proof. By virtue of Remark 4.7.4.7, we may assume that $\kappa $ is a regular cardinal, In particular, $\kappa $ has cofinality larger than $\aleph _0$. It will therefore suffice to prove that $\operatorname{Ex}^{n}(S)$ is $\kappa $-small, for each integer $n \geq 0$. By virtue of Proposition 4.7.4.10, it will suffice to show that for every finite simplicial set $K$, the set $\operatorname{Hom}_{\operatorname{Set_{\Delta }}}( K, \operatorname{Ex}^{n}(S) ) \simeq \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{Sd}^{n}(K), S )$ is $\kappa $-small. This follows from Proposition 4.7.4.10, since $S$ is $\kappa $-small and $\operatorname{Sd}^{n}(K)$ is finite (see Remark 3.3.3.6). $\square$