Kerodon

Example 7.3.3.9. Let $\overline{F}: \operatorname{\mathcal{C}}^{\triangleright } \rightarrow \operatorname{\mathcal{D}}$ and $U: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be functors of $\infty $-categories, and set $F = \overline{F}|_{\operatorname{\mathcal{C}}}$. Then $F$ is $U$-left Kan extended from a full subcategory $\operatorname{\mathcal{C}}^{0} \subseteq \operatorname{\mathcal{C}}$ if and only if $\overline{F}$ is $U$-left Kan extended from the cone $(\operatorname{\mathcal{C}}^0)^{\triangleright } \subseteq \operatorname{\mathcal{C}}^{\triangleright }$. To prove this, it suffices (by virtue of Proposition 7.3.3.7) to show that $F$ is $U$-left Kan extended from $\operatorname{\mathcal{C}}^{0}$ at an object $C \in \operatorname{\mathcal{C}}$ if and only if $\overline{F}$ is $U$-left Kan extended from $( \operatorname{\mathcal{C}}^{0} )^{\triangleright }$ at $C$, which follows immediately from the definition.