Kerodon

Proof. Since $\kappa \trianglelefteq \kappa ^{+}$ (see Example 9.1.7.7), we can use Theorem 9.1.8.7 to reduce to the case where $\operatorname{\mathcal{C}}= \operatorname{N}_{\bullet }(A)$, where $(A, \leq )$ is a $\kappa $-directed partially ordered set of cardinality $\leq \kappa $. Choose a surjective map of sets $f: \mathrm{Ord}_{< \kappa } \rightarrow A$. Let $Q$ denote the collection of all pairs $(\alpha , g_{< \alpha } )$, where $\alpha \leq \kappa $ is an ordinal and $g_{< \alpha }: \mathrm{Ord}_{< \alpha } \rightarrow A$ is a nondecreasing function satisfying $g_{< \alpha }(\alpha _0) \geq f(\alpha _0)$ for each $\alpha _0 < \alpha $. We regard $Q$ as a partially ordered set, where $(\alpha , g_{< \alpha }) \leq (\beta , g_{< \beta } )$ if $\alpha \leq \beta $ and $g_{<\alpha } = g_{< \beta }|_{ \mathrm{Ord}_{< \alpha } }$. The partially ordered set $Q$ satisfies the hypotheses of Zorn's lemma, and therefore has a maximal element $(\gamma , g_{< \gamma } )$. If $\gamma < \kappa $, then $\{ f(\gamma ) \} \cup \{ g_{< \gamma }(\alpha ): \alpha < \gamma \} $ is a $\kappa $-small subset of $A$, and therefore admits an upper bound $a \in A$. In this case, we can extend $g_{< \gamma }$ to a nondecreasing function $g_{\leq \gamma }: \mathrm{Ord}_{\leq \gamma } \rightarrow A$ by setting $g_{\leq \gamma }(\gamma ) = a$, contradicting the maximality of the pair $(\gamma , g_{< \gamma } )$. It follows that $\gamma = \kappa $: that is, we can regard $g_{< \gamma }$ as a nondecreasing function from $\mathrm{Ord}_{< \kappa } \rightarrow A$. Using Example 9.1.4.10, we see that the induced map $\operatorname{N}_{\bullet }(\mathrm{Ord}_{< \kappa }) \rightarrow \operatorname{N}_{\bullet }(A)$ is a right cofinal functor of $\infty $-categories. $\square$