Kerodon

Remark 7.3.7.12. In the situation of Lemma 7.3.7.10, suppose we are given an inner anodyne morphism of simplicial sets $A \hookrightarrow B$. Then every diagram $K^{\triangleright } \star A \rightarrow \operatorname{\mathcal{E}}$ can be extended to a morphism $K^{\triangleright } \star B \rightarrow \operatorname{\mathcal{E}}$, so that condition $(\ast _ A)$ is satisfied if and only if condition $(\ast _ B)$ is satisfied. Consequently, to show that every object of $\operatorname{Fun}'_{/\operatorname{\mathcal{E}}}( K^{\triangleright }, \operatorname{\mathcal{D}})$ is a $V$-colimit diagram, it suffices to verify condition $(\ast _{A})$ in the special case where $A$ is an $\infty $-category (see Corollary 4.1.3.3).