Remark 7.4.1.2. In the situation of Theorem 7.4.1.1, the restriction map $\operatorname{Fun}_{ / \operatorname{\mathcal{C}}^{\triangleleft } }^{\operatorname{CCart}}( \operatorname{\mathcal{C}}^{\triangleleft }, \overline{\operatorname{\mathcal{E}}} ) \rightarrow \operatorname{Fun}_{ / \operatorname{\mathcal{C}}}^{\operatorname{CCart}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$ is automatically an isofibration of $\infty $-categories (Remark 5.3.1.18). Using Proposition 4.5.5.20, we see that condition $(1)$ of Theorem 7.4.1.1 is equivalent to the following a priori stronger condition:
- $(1')$
The restriction map
\[ \operatorname{Fun}_{ / \operatorname{\mathcal{C}}^{\triangleleft } }^{\operatorname{CCart}}( \operatorname{\mathcal{C}}^{\triangleleft }, \overline{\operatorname{\mathcal{E}}} ) \rightarrow \operatorname{Fun}_{ / \operatorname{\mathcal{C}}}^{\operatorname{CCart}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}}) \]is a trivial Kan fibration of simplicial sets.