Kerodon

Remark 7.3.3.20. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ and $U: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be functors of $\infty $-categories, and let $\operatorname{\mathcal{C}}^{0} \subseteq \operatorname{\mathcal{C}}$ be a full subcategory. Suppose that $U \circ F$ is left Kan extended from $\operatorname{\mathcal{C}}^{0}$. Then $F$ is left Kan extended from $\operatorname{\mathcal{C}}^{0}$ if and only if it is $U$-left Kan extended from $\operatorname{\mathcal{C}}^{0}$; this follows by applying Remark 7.3.3.19 in the special case $\operatorname{\mathcal{E}}' = \Delta ^0$. Similarly, if $U \circ F$ is right Kan extended from $\operatorname{\mathcal{C}}^{0}$, then $F$ is right Kan extended from $\operatorname{\mathcal{C}}^{0}$ if and only if it is $U$-right Kan extended from $\operatorname{\mathcal{C}}^{0}$.