Kerodon

Proof. Fix a set $S$ of cardinality $\kappa $. Proposition 4.7.4.10 implies that a simplicial set $X_{\bullet }$ is $\kappa ^{+}$-small if and only if, for each $n \geq 0$, the set of $n$-simplices $X_{n}$ has cardinality $\leq \kappa $. In this case, $X_{\bullet }$ is isomorphic to a simplicial set $Y_{\bullet }$, where each $Y_{n}$ is given as a subset of $S$. Such a simplicial set $Y_{\bullet }$ is determined by the following data:

• For each $n \geq 0$, the specification of the subset $Y_{n} \subseteq S$.

• For every nondecreasing function $\alpha : [m] \rightarrow [n]$, the specification of a function $\alpha ^{\ast }: Y_{n} \rightarrow Y_{m} \subseteq S$; in this case, we can assume that $\alpha ^{\ast }$ is the restriction of a function $\overline{\alpha }^{\ast }: S \rightarrow S$.

In particular, $Y_{\bullet }$ can be described by specifying a countable collection of subsets of $S$, together with a countable collection of functions from $S$ to itself. The collection of all such specifications has cardinality $\leq 2^{\kappa }$. $\square$