Example 5.1.7.7. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $F: K \rightarrow \operatorname{\mathcal{C}}$ be a diagram. It follows from Theorem 4.6.4.17 and Proposition 5.1.7.5 that the slice and coslice diagonal morphisms
\[ \delta _{/F}: \operatorname{\mathcal{C}}_{/F} \hookrightarrow \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{ \operatorname{Fun}(K, \operatorname{\mathcal{C}}) } \{ F\} \quad \quad \delta _{F/}: \operatorname{\mathcal{C}}_{F/} \hookrightarrow \{ F\} \operatorname{\vec{\times }}_{ \operatorname{Fun}(K, \operatorname{\mathcal{C}}) } \operatorname{\mathcal{C}} \]
are equivalences of right and left fibrations over $\operatorname{\mathcal{C}}$, respectively. In particular, for every morphism of simplicial sets $\operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$, the induced maps
\[ \operatorname{\mathcal{D}}\times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{/F} \hookrightarrow \operatorname{\mathcal{D}}\operatorname{\vec{\times }}_{ \operatorname{Fun}(K,\operatorname{\mathcal{C}}) } \{ F\} \quad \quad \operatorname{\mathcal{D}}\times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{F/} \hookrightarrow \{ F\} \operatorname{\vec{\times }}_{ \operatorname{Fun}(K, \operatorname{\mathcal{C}}) } \operatorname{\mathcal{D}} \]
are equivalences of inner fibrations over $\operatorname{\mathcal{D}}$ (Remark 5.1.7.4); in particular, they are categorical equivalences of simplicial sets (Proposition 5.1.7.5).