Corollary 7.7.2.33. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category which admits pullbacks and let $K$ be a simplicial set. The following conditions are equivalent:
- $(1)$
Every weak colimit diagram $F: K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$ is a descent diagram for the evaluation functor $\operatorname{ev}_{1}: \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}$.
- $(2)$
For every morphism $u: C' \rightarrow C$ of $\operatorname{\mathcal{C}}$, the pullback functor
\[ u^{\ast }: \operatorname{\mathcal{C}}_{/C} \rightarrow \operatorname{\mathcal{C}}_{/C'} \quad \quad X \mapsto C' \times _{C} X \]preserves $K$-indexed colimits.