Kerodon

Proof. The implication $(1) \Rightarrow (2)$ is trivial and the implication $(2) \Rightarrow (3)$ follows from Corollary 9.4.7.18. If condition $(3)$ is satisfied, then we can choose a small regular cardinal $\lambda $ for which the $\infty $-category $\operatorname{\mathcal{C}}$ is $\lambda $-accessible and $(\iota \circ L)$ is $\lambda $-finitary (Remark 9.4.7.2). It then follows that $\operatorname{\mathcal{C}}'$ is closed under small $\lambda $-filtered colimits, which proves that $(3) \Rightarrow (4)$. It will therefore suffice to show that $(4)$ implies $(1)$.

Fix a small regular cardinal $\kappa $ such that $\operatorname{\mathcal{C}}$ is $\kappa $-accessible and the full subcategory $\operatorname{\mathcal{C}}' \subseteq \operatorname{\mathcal{C}}$ is closed under small $\kappa $-filtered colimits. Then $\operatorname{\mathcal{C}}'$ admits small $\kappa $-filtered colimits which are preserved by the inclusion functor $\iota $. Consequently, to prove $(1)$, it will suffice to show that the $\infty $-category $\operatorname{\mathcal{C}}'$ is accessible. In fact, we will show that $\operatorname{\mathcal{C}}'$ is $\kappa $-accessible. Let $\operatorname{\mathcal{C}}_{< \kappa }$ be the full subcategory of $\operatorname{\mathcal{C}}$ spanned by the $\kappa $-compact objects, and let $\operatorname{\mathcal{C}}'_0 \subseteq \operatorname{\mathcal{C}}'$ be the essential image of the functor $L|_{ \operatorname{\mathcal{C}}_{< \kappa } }$. Since the $\infty $-category $\operatorname{\mathcal{C}}_{< \kappa }$ is essentially small (Proposition 9.4.6.2) and $\operatorname{\mathcal{C}}' \subseteq \operatorname{\mathcal{C}}$ is locally small (Remark 9.4.6.14), the full subcategory $\operatorname{\mathcal{C}}'_0 \subseteq \operatorname{\mathcal{C}}'$ is essentially small. Our assumption that $\iota $ is $\kappa $-finitary, guarantees that the functor $L$ is $\kappa $-compact (Example 9.4.2.17), so that each object of $\operatorname{\mathcal{C}}'_0$ is $\kappa $-compact when viewed as an object of $\operatorname{\mathcal{C}}'$. Since $\operatorname{\mathcal{C}}$ is $\kappa $-accessible, every object $X \in \operatorname{\mathcal{C}}$ can be realized as the colimit of a small $\kappa $-filtered diagram in $\operatorname{\mathcal{C}}_{< \kappa }$, so that $L(X) \in \operatorname{\mathcal{C}}'$ can be realized as the colimit of a small $\kappa $-filtered diagram in $\operatorname{\mathcal{C}}'_0$. It follows that $\operatorname{\mathcal{C}}'$ is an $\operatorname{Ind}_{\kappa }$-completion of the essentially small subcategory $\operatorname{\mathcal{C}}'_0$, and therefore $\kappa $-accessible. $\square$