Kerodon

Remark 7.3.3.19 (Transitivity). Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$, $U: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$, and $V: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{E}}'$ 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 $V$-left Kan extended from $\operatorname{\mathcal{C}}^{0}$. Then $F$ is $U$-left Kan extended from $\operatorname{\mathcal{C}}^{0}$ if and only if it is $(V \circ U)$-left Kan extended from $\operatorname{\mathcal{C}}^{0}$ (see Proposition 7.1.5.14). Similarly, if $U \circ F$ is $V$-right Kan extended from $\operatorname{\mathcal{C}}^{0}$, then $F$ is $U$-right Kan extended from $\operatorname{\mathcal{C}}^{0}$ if and only if it is $(V \circ U)$-right Kan extended from $\operatorname{\mathcal{C}}^{0}$.