Corollary 7.7.2.23. Let $K$ be a simplicial set and let $\operatorname{\mathcal{C}}$ be an $\infty $-category which admits pullbacks and $K$-indexed colimits. Then $K$-indexed colimits in $\operatorname{\mathcal{C}}$ are strongly universal if and only if the following condition is satisfied:
- $(\ast )$
Let $\gamma : G \rightarrow F$ be a natural transformation between diagrams $G,F: K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$, where $F$ is a colimit diagram and $\gamma |_{K}$ is cartesian. Then $\gamma $ is cartesian if and only if $G$ is a colimit diagram.