Proof.
Let $\operatorname{\mathcal{E}}\subseteq \operatorname{\mathcal{QC}}_{< \lambda }$ be the smallest full subcategory which contains $\Delta ^1$ and is closed under $\kappa $-filtered colimits. Applying Proposition 9.2.8.8, we conclude that every essentially finite $\infty $-category belongs to $\operatorname{\mathcal{E}}$. Let $Q: \operatorname{Set_{\Delta }}\rightarrow \operatorname{Set_{\Delta }}$ be as in the proof of Proposition 9.2.8.13. 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 9.2.2.21. We will complete the proof by showing that $(3)$ implies $(1)$. Combining Variant 9.1.7.21 with Corollary 9.1.6.3, we see that $\operatorname{\mathcal{QC}}_{< \lambda }$ is generated by $\operatorname{\mathcal{QC}}_{< \kappa }$ under $\lambda $-small, $\kappa $-filtered colimits. Applying Lemma 9.2.6.12, we conclude that if $\operatorname{\mathcal{C}}$ is $(\kappa ,\lambda )$-compact, then it is a retract of a $\kappa $-small $\infty $-category $\operatorname{\mathcal{D}}$. In particular, it can be realized as the colimit of a diagram
\[ \operatorname{\mathcal{D}}\xrightarrow {F} \operatorname{\mathcal{D}}\xrightarrow {F} \operatorname{\mathcal{D}}\xrightarrow {F} \operatorname{\mathcal{D}}\rightarrow \cdots \]
in the $\infty $-category $\operatorname{\mathcal{QC}}_{<\lambda }$, for some (homotopy idempotent) functor $F: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{D}}$. Forming the colimit of this diagram in the ordinary category of simplicial sets, we obtain a $\kappa $-small $\infty $-category which is equivalent to $\operatorname{\mathcal{C}}$ (see Example 7.6.5.8).
$\square$