Kerodon

Proof. If $Q$ is small, then the $\infty $-category $\operatorname{\mathcal{C}}$ is essentially small and idempotent-complete (Example 8.5.4.6), hence accessible by virtue of Proposition 9.4.6.21. For the converse, suppose that $\operatorname{\mathcal{C}}$ is accessible. Then $\operatorname{\mathcal{C}}$ can be identified with the $\operatorname{Ind}_{\kappa }$-completion of a full subcategory $\operatorname{\mathcal{C}}_0 \subseteq \operatorname{\mathcal{C}}$, for some small regular cardinal $\kappa $. Let us identify $\operatorname{\mathcal{C}}_0$ with the nerve of a partially ordered set $Q_0 \subseteq Q$. It follows from Exercise 9.3.2.12 that $\operatorname{Ind}(\operatorname{\mathcal{C}}_0)$ can be identified with the nerve of the partially ordered set of ideals in $Q_0$. In particular, $\operatorname{Ind}(\operatorname{\mathcal{C}}_0)$ is essentially small, so that $\operatorname{\mathcal{C}}\simeq \operatorname{Ind}_{\kappa }(\operatorname{\mathcal{C}}_0)$ is also essentially small. $\square$