Remark 7.1.7.11. In the situation of Definition 7.1.7.9, we can reformulate condition $(\ast )$ as follows:
- $(\ast ')$
Let $e: C \rightarrow C'$ of the simplicial set $\operatorname{\mathcal{C}}$, let $X$ be an object of the $\infty $-category $\operatorname{\mathcal{E}}_{C'}$, and set $\operatorname{\mathcal{E}}' = \Delta ^1 \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$. Then the restriction map
\[ \theta : \operatorname{Hom}_{ \operatorname{Fun}( K^{\triangleleft }, \operatorname{\mathcal{E}}' ) }(\overline{q}, \underline{X}) \rightarrow \operatorname{Hom}_{ \operatorname{Fun}( K, \operatorname{\mathcal{E}}' ) }( \overline{q}_{K}, \underline{X}|_{K} ) \]is a homotopy equivalence of Kan complexes, where $\underline{X}$ denotes the constant diagram $K^{\triangleright } \rightarrow \operatorname{\mathcal{E}}'$ taking the value $X$.
The implication $(\ast ) \Rightarrow (\ast ')$ follows from Corollary 7.1.6.20. To prove the converse, it suffices to show that the morphism $\theta $ is also a homotopy equivalence when $X$ belongs to $\operatorname{\mathcal{E}}_{C}$, which follows by applying condition $(\ast ')$ in the special case $e = \operatorname{id}_{C}$.