Corollary 5.3.7.5. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $K$ be a simplicial set. Then:
- $(1)$
The restriction map $U: \operatorname{Fun}(K^{\triangleleft }, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}(K, \operatorname{\mathcal{C}})$ is a cocartesian fibration. Moreover, a morphism $e$ of $\operatorname{Fun}(K^{\triangleleft }, \operatorname{\mathcal{C}})$ is $U$-cocartesian if and only if it carries the cone point ${\bf 0} \in K^{\triangleleft }$ to an isomorphism in $\operatorname{\mathcal{C}}$.
- $(2)$
The restriction map $V: \operatorname{Fun}(K^{\triangleright }, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}(K, \operatorname{\mathcal{C}})$ is a cartesian fibration. Moreover, a morphism $e$ of $\operatorname{Fun}(K^{\triangleright }, \operatorname{\mathcal{C}})$ is $U$-cartesian if and only if it carries the cone point ${\bf 1} \in K^{\triangleright }$ to an isomorphism in $\operatorname{\mathcal{C}}$.