Corollary 7.7.1.20. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $K$ be a simplicial set. The following conditions are equivalent:
Every colimit diagram $\overline{\mathscr {F}}: K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$ is a universal colimit diagram.
In the $\infty $-category $\operatorname{\mathcal{C}}$, $K$-indexed colimits are universal.
Proof. The implication $(2) \Rightarrow (1)$ follows from Proposition 7.7.1.19. The reverse implication follows from the observation that, if $\mathscr {F}: K \rightarrow \operatorname{\mathcal{C}}$ is a diagram, then every natural transformation from $\mathscr {F}$ to a constant diagram is homotopic to one which arises from a diagram $\overline{\mathscr {F}}: K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$ extending $\mathscr {F}$ (see Theorem 4.6.4.17). $\square$