Kerodon

Definition 7.7.2.13. Let $K$ be a simplicial set and let $\operatorname{\mathcal{C}}$ be an $\infty $-category which admits pullbacks and $K$-indexed colimits. We say that a morphism $F: K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$ is a strongly universal colimit diagram if the restriction map $\operatorname{Fun}( K^{\triangleright }, \operatorname{\mathcal{C}})^{\operatorname{Cart}}_{/F} \rightarrow \operatorname{Fun}( K, \operatorname{\mathcal{C}})^{\operatorname{Cart}}_{/F|_{K}}$ is an equivalence of $\infty $-categories: that is, if $F$ is an effective descent diagram for the evaluation functor $\operatorname{ev}_{1}: \operatorname{Fun}(\Delta ^1, \operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}$ (Remark 7.7.2.11).