Corollary 9.2.9.2. Let $\mu $ be an uncountable cardinal and let $\operatorname{\mathcal{D}}$ be the limit of a finite diagram
\[ \mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{QC}}_{< \mu } \quad \quad (C \in \operatorname{\mathcal{C}}) \mapsto \mathscr {F}(C) \]
satisfying the following conditions:
- $(1)$
For each vertex $C \in \operatorname{\mathcal{C}}$, the $\infty $-category $\mathscr {F}(C)$ admits small filtered colimits.
- $(2)$
For each edge $e: C \rightarrow C'$ of $\operatorname{\mathcal{C}}$, the functor $\mathscr {F}(e): \mathscr {F}(C) \rightarrow \mathscr {F}(C')$ is finitary.
Let $D$ be an object of $\operatorname{\mathcal{D}}$. If the image of $D$ in each $\mathscr {F}(C)$ is compact, then $D$ is compact.