Definition 7.7.2.15. Let $K$ be a simplicial set and let $\operatorname{\mathcal{C}}$ be an $\infty $-category which admits pullbacks and $K$-indexed colimits. Then, for every diagram $F: K \rightarrow \operatorname{\mathcal{C}}$, we can choose a colimit diagram $\overline{F}: K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$ satisfying $\overline{F}|_{K} = F$. We say that the colimit of $F$ is strongly universal if $\overline{F}$ is a strongly universal colimit diagram, in the sense of Definition 7.7.2.13. By virtue of Remark 7.7.2.4, this condition depends only the diagram $F$ (and not on the choice of extension $\overline{F}: K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$).
We say that $K$-indexed colimits in $\operatorname{\mathcal{C}}$ are strongly universal if, for every diagram $F: K \rightarrow \operatorname{\mathcal{C}}$, the colimit of $F$ is strongly universal.