Kerodon

Proof. The $\infty $-category $\operatorname{\mathcal{QC}}$ admits small filtered colimits (Corollary 7.4.5.3), and every object of $\operatorname{\mathcal{QC}}_{\mathrm{fin}}$ is compact when viewed as an object of $\operatorname{\mathcal{QC}}$ (Proposition 9.2.8.9). By virtue of Corollary 9.2.3.6, it will suffice to show that every small $\infty $-category $\operatorname{\mathcal{C}}$ can be realized as the colimit of a small filtered diagram $\{ \operatorname{\mathcal{C}}_{\alpha } \} $ in the $\infty $-category $\operatorname{\mathcal{QC}}$, where each $\operatorname{\mathcal{C}}_{\alpha }$ is essentially finite. Let $Q: \operatorname{Set_{\Delta }}\rightarrow \operatorname{Set_{\Delta }}$ be a finitary functor which associates to each simplicial set $K$ an $\infty $-category $Q(K)$ equipped with a categorical equivalence $u_ K: K \rightarrow Q(K)$ (Proposition 4.1.3.2). Without loss of generality, we may assume that $\operatorname{\mathcal{C}}= Q(K)$ for some small simplicial set $K$. In this case, we can take $\{ \operatorname{\mathcal{C}}_{\alpha } \} $ to be the diagram $\{ Q( K_{\alpha } ) \} $, where $K_{\alpha }$ ranges over the collection fo all finite simplicial subsets of $K$ (see Remark 3.6.1.8). $\square$