Kerodon

Corollary 7.3.8.4. 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 we are given a right fibration of $\infty $-categories $V: \operatorname{\mathcal{B}}\rightarrow \operatorname{\mathcal{C}}$ and set $\operatorname{\mathcal{B}}^{0} = \operatorname{\mathcal{C}}^{0} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{B}}$. If $F$ is $U$-left Kan extended from $\operatorname{\mathcal{C}}^{0}$, then $F \circ V$ is $U$-left Kan extended from $\operatorname{\mathcal{B}}^{0}$. The converse holds if every fiber of $V$ is nonempty.