Proof.
Let $\operatorname{\mathcal{E}}\subseteq \operatorname{\mathcal{QC}}_{< \lambda }$ be the smallest full subcategory which contains $\Delta ^1$ and is closed under $\kappa $-small colimits. Applying Proposition 9.2.7.8, we conclude that every essentially finite $\infty $-category belongs to $\operatorname{\mathcal{E}}$. Let $Q: \operatorname{Set_{\Delta }}\rightarrow \operatorname{QCat}$ be as in the proof of Proposition 9.2.7.12. If $\operatorname{\mathcal{C}}$ is a $\kappa $-small simplicial set, then it can be realized as the colimit of a $\kappa $-small filtered diagram $\{ \operatorname{\mathcal{C}}_{\alpha } \} $ of finite simplicial sets $\operatorname{\mathcal{C}}_{\alpha }$, so that $Q( \operatorname{\mathcal{C}})$ is the colimit of a $\kappa $-small filtered diagram $\{ Q(\operatorname{\mathcal{C}}_{\alpha } ) \} $ in the $\infty $-category $\operatorname{\mathcal{QC}}_{< \lambda }$. Since each $Q(\operatorname{\mathcal{C}}_{\alpha } )$ is essentially finite, it follows that $Q( \operatorname{\mathcal{C}})$ belongs to $\operatorname{\mathcal{E}}$. If $\operatorname{\mathcal{C}}$ is an $\infty $-category, then it is equivalent to $Q(\operatorname{\mathcal{C}})$ and therefore also belongs to $\operatorname{\mathcal{E}}$. This establishes the implication $(1) \Rightarrow (2)$.
The implication $(2) \Rightarrow (3)$ follows from Proposition 9.2.5.24. We will complete the proof by showing that $(3)$ implies $(1)$. Using Variant 9.1.7.21, we can write $\operatorname{\mathcal{C}}$ as the colimit of a diagram
\[ (A, \leq ) \rightarrow \operatorname{QCat}\quad \quad \alpha \mapsto \operatorname{\mathcal{C}}_{\alpha }, \]
where $(A, \leq )$ is a $\lambda $-small $\kappa $-directed partially ordered set and each $\operatorname{\mathcal{C}}_{\alpha }$ is a $\kappa $-small $\infty $-category. Corollary 9.1.6.3 implies that $\operatorname{\mathcal{C}}$ is the also the colimit of the diagram $\{ \operatorname{\mathcal{C}}_{\alpha } \} $ in the $\infty $-category $\operatorname{\mathcal{QC}}_{< \lambda }$. If $\operatorname{\mathcal{C}}$ is a $(\kappa ,\lambda )$-compact object of $\operatorname{\mathcal{QC}}_{< \lambda }$, it follows that the identity functor $\operatorname{id}_{\operatorname{\mathcal{C}}}$ factors up to homotopy through some $\operatorname{\mathcal{C}}_{\alpha }$. In particular, $\operatorname{\mathcal{C}}$ is a retract of $\operatorname{\mathcal{C}}_{\alpha }$, and can therefore be realized as the colimit of a diagram
\[ \operatorname{\mathcal{C}}_{\alpha } \xrightarrow {e} \operatorname{\mathcal{C}}_{\alpha } \xrightarrow {e} \operatorname{\mathcal{C}}_{\alpha } \xrightarrow {e} \operatorname{\mathcal{C}}_{\alpha } \rightarrow \cdots \]
in the $\infty $-category $\operatorname{\mathcal{S}}_{< \lambda }$, for some (homotopy idempotent) endomorphism $e$ of the $\infty $-category $\operatorname{\mathcal{C}}_{\alpha }$. Forming the colimit of this diagram in the ordinary category of simplicial sets, we obtain a $\kappa $-small $\infty $-category which is equivalent to $\operatorname{\mathcal{C}}$ (Corollary 9.1.6.3).
$\square$