Corollary 7.7.2.12. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category which admits pullbacks and let $K$ be a simplicial set. If $K$-indexed colimits in $\operatorname{\mathcal{C}}$ are universal, then the restriction functor
\[ \operatorname{Fun}( K^{\triangleright }, \operatorname{\mathcal{C}})^{\operatorname{Cart}}_{/F} \rightarrow \operatorname{Fun}( K, \operatorname{\mathcal{C}})^{\operatorname{Cart}}_{/F|_{K}} \]
for every colimit diagram $F: K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$. The converse holds $\operatorname{\mathcal{C}}$ admits $K$-indexed colimits or has a final object.