Kerodon

Remark 7.3.2.4. Let $F: \operatorname{\mathcal{B}}\star \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories and set $G = F|_{\operatorname{\mathcal{B}}}$, so that $F$ can be identified with a functor $f: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}_{G/}$. If $\operatorname{\mathcal{C}}^{0}$ is a full subcategory of $\operatorname{\mathcal{C}}$, then $f$ is left Kan extended from $\operatorname{\mathcal{C}}^{0}$ at an object $C \in \operatorname{\mathcal{C}}$ if and only if $F$ is left Kan extended from $\operatorname{\mathcal{B}}\star \operatorname{\mathcal{C}}^{0}$ at $C$ (see Remark 7.1.3.11). Combining this observation with Exercise 7.3.2.3, we see that $f$ is left Kan extended from $\operatorname{\mathcal{C}}^{0}$ if and only if $F$ is left Kan extended from $\operatorname{\mathcal{B}}\star \operatorname{\mathcal{C}}^{0}$.