Proof.
If $X$ is essentially $\kappa $-small, then there exists a weak homotopy equivalence $K \rightarrow X$, where $K$ is a $\kappa $-small simplicial set. In this case, $X$ can be viewed as a colimit of the constant diagram $K \rightarrow \{ \Delta ^0 \} \hookrightarrow \operatorname{\mathcal{S}}_{< \lambda }$ (Example 7.1.2.9). This proves the implication $(1) \Rightarrow (2)$. Note that the object $\Delta ^0 \in \operatorname{\mathcal{S}}_{< \lambda }$ corepresents the identity functor $\operatorname{id}: \operatorname{\mathcal{S}}_{< \lambda } \rightarrow \operatorname{\mathcal{S}}_{< \lambda }$, and is therefore $(\kappa ,\lambda )$-compact. Consequently, the implication $(2) \Rightarrow (3)$ follows from Proposition 9.2.2.21.
We will complete the proof by showing that $(3)$ implies $(1)$. Suppose that $X$ is $(\kappa ,\lambda )$-compact when viewed as an object of the $\infty $-category $\operatorname{\mathcal{S}}_{< \lambda }$; we wish to show that $X$ is essentially $\kappa $-small. Without loss of generality we may assume that $X$ is $\lambda $-small. Using our assumption that $\kappa \trianglelefteq \lambda $, we can choose a $\lambda $-small collection of $\kappa $-small simplicial subsets $\{ X_{\alpha } \} _{\alpha \in A}$ such that every $\kappa $-small simplicial subset of $X$ is contained in some $X_{\alpha }$ (Lemma 9.1.7.18). We regard $A$ as a partially ordered set, with $\alpha \leq \beta $ if $X_{\alpha }$ is contained in $X_{\beta }$. The partially ordered set $(A, \leq )$ is then $\kappa $-directed (Remark 9.1.7.19). Let $\operatorname{Ex}^{\infty }: \operatorname{Set_{\Delta }}\rightarrow \operatorname{Set_{\Delta }}$ be the functor defined in Construction 3.3.6.1. Since $\operatorname{Ex}^{\infty }$ preserves filtered colimits (Proposition 3.3.6.4), we can identify $\operatorname{Ex}^{\infty }(X)$ with the colimit of the diagram $\{ \operatorname{Ex}^{\infty }( X_{\alpha } ) \} _{\alpha \in A}$ in the category of Kan complexes, and therefore also in the $\infty $-category $\operatorname{\mathcal{S}}_{< \lambda }$ (Variant 9.1.6.4). Note that each $\operatorname{Ex}^{\infty }(X_{\alpha } )$ is a $\kappa $-small Kan complex, and therefore $(\kappa ,\lambda )$-compact as an object of the $\infty $-category $\operatorname{\mathcal{S}}_{< \lambda }$. Applying Lemma 9.2.6.12, we deduce that there exist some index $\alpha \in A$ such that $X$ is a retract of $\operatorname{Ex}^{\infty }(X_{\alpha } )$ in the $\infty $-category $\operatorname{\mathcal{S}}_{< \lambda }$. In particular, $X$ can be realized as the colimit of a diagram
\[ \operatorname{Ex}^{\infty }(X_{\alpha } ) \xrightarrow {e} \operatorname{Ex}^{\infty }(X_{\alpha } ) \xrightarrow {e} \operatorname{Ex}^{\infty }(X_{\alpha } ) \xrightarrow {e} \operatorname{Ex}^{\infty }(X_{\alpha } ) \rightarrow \cdots \]
in the $\infty $-category $\operatorname{\mathcal{S}}_{< \lambda }$, for some (homotopy idempotent) endomorphism $e$ of the Kan complex $\operatorname{Ex}^{\infty }( X_{\alpha } )$. Forming the colimit of this diagram in the ordinary category of simplicial sets, we obtain a $\kappa $-small Kan complex which is homotopy equivalent to $X$ (Variant 7.6.5.9).
$\square$