Corollary 7.1.7.7. Let $K$ be a simplicial set, let $\operatorname{\mathcal{C}}$ be an $\infty $-category which admits $K$-indexed colimits, and let $\operatorname{Fun}'( K^{\triangleright }, \operatorname{\mathcal{C}})$ denote the full subcategory of $\operatorname{Fun}( K^{\triangleright }, \operatorname{\mathcal{C}})$ spanned by the colimit diagrams. Then the restriction map
\[ \operatorname{Fun}'( K^{\triangleright }, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}( K, \operatorname{\mathcal{C}}) \quad \quad \overline{f} \mapsto \overline{f}|_{K} \]
is a trivial Kan fibration.