Proposition 9.2.8.9. Let $\operatorname{\mathcal{C}}$ be an essentially finite $\infty $-category. Then $\operatorname{\mathcal{C}}$ is compact when regarded as an object of the $\infty $-category $\operatorname{\mathcal{QC}}$.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. Let $h^{\operatorname{\mathcal{C}}}: \operatorname{\mathcal{QC}}\rightarrow \operatorname{\mathcal{S}}$ be the functor corepresented by $\operatorname{\mathcal{C}}$; we wish to show that the functor $h^{\operatorname{\mathcal{C}}}$ is finitary. By virtue of Corollary 9.1.9.13, it will suffice to show that the functor $h^{\operatorname{\mathcal{C}}}$ preserves the colimit of every diagram $\mathscr {F}: \operatorname{N}_{\bullet }(A) \rightarrow \operatorname{\mathcal{QC}}$ indexed by a small directed partially ordered set $(A,\leq )$. Using Corollary 5.6.5.18, we can reduce to the case where $\mathscr {F}$ factors through the simplicial subset $\operatorname{N}_{\bullet }(\operatorname{QCat}) \subset \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{QCat}) = \operatorname{\mathcal{QC}}$: that is, it is given by a diagram in the ordinary category of simplicial sets.
Choose a categorical equivalence $f: K \rightarrow \operatorname{\mathcal{C}}$, where $K$ is a finite simplicial set. Using Proposition 5.6.6.17, we can assume that $h^{\operatorname{\mathcal{C}}}$ is obtained as the homotopy coherent nerve of the simplicially enriched functor $\operatorname{Fun}(K, \bullet )^{\simeq }: \operatorname{QCat}\rightarrow \operatorname{Kan}$. In particular, we have a commutative diagram of $\infty $-categories
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{N}_{\bullet }(\operatorname{QCat}) \ar [r]^{\operatorname{Fun}(K, \bullet )^{\simeq }} \ar [d] & \operatorname{N}_{\bullet }(\operatorname{Kan}) \ar [d] \\ \operatorname{\mathcal{QC}}\ar [r]^{ h^{\operatorname{\mathcal{C}}} } & \operatorname{\mathcal{S}}. } \]
Consequently, to show that the functor $h^{\operatorname{\mathcal{C}}}$ preserves the colimit of the diagram $\mathscr {F}$, it will suffice to show that the vertical maps and upper horizontal map in this diagram preserve $\operatorname{N}_{\bullet }(A)$-indexed colimits. For the vertical maps, this follows from Corollary 9.1.6.3 and Variant 9.1.6.4. For the upper horizontal map, it follows from Variant 9.2.0.5. $\square$