Corollary 7.7.2.34. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category which admits pullbacks and let $K$ be a simplicial set. Assume either that $\operatorname{\mathcal{C}}$ has a final object or that $\operatorname{\mathcal{C}}$ admits $K$-indexed colimits. Then the following conditions are equivalent:
- $(1)$
Every colimit diagram $F: K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$ is a universal colimit diagram.
- $(2)$
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.