Corollary 7.3.8.11. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ and $U: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be functors of $\infty $-categories, where $U$ is an inner fibration and $F$ is $U$-left Kan extended from a full subcategory $\operatorname{\mathcal{C}}^{0} \subseteq \operatorname{\mathcal{C}}$. Set $F_0 = F|_{ \operatorname{\mathcal{C}}^{0} }$. Then the restriction map
\[ \theta : \operatorname{\mathcal{D}}_{ F / } \rightarrow \operatorname{\mathcal{D}}_{ F_0 / } \times _{ \operatorname{\mathcal{E}}_{ (U \circ F_0) / } } \operatorname{\mathcal{E}}_{ (U \circ F) / } \]
is a trivial Kan fibration.