Remark 7.1.6.25. In the situation of Proposition 7.1.6.24, we can replace condition $(\ast )$ by the following:
- $(\ast ')$
For every morphism $e: D \rightarrow D'$ in $\operatorname{\mathcal{D}}$, the morphism
\[ K^{\triangleleft } \xrightarrow { \overline{f} } \operatorname{\mathcal{C}}_{D} \hookrightarrow \Delta ^{1} \times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{C}} \]is a colimit diagram in the $\infty $-category $\Delta ^1 \times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{C}}$.
See Corollary 7.1.6.18.