Corollary 8.5.1.9 (03YR)

Corollary 8.5.1.9. 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 $F$ is $U$-left Kan extended from $\operatorname{\mathcal{C}}^{0}$ at an object $X \in \operatorname{\mathcal{C}}$. If $Y \in \operatorname{\mathcal{C}}$ is a retract of $X$, then $F$ is also $U$-left Kan extended from $\operatorname{\mathcal{C}}^{0}$ at $Y$.

Proof. Without loss of generality, we may assume that $\operatorname{\mathcal{C}}$ is spanned by $\operatorname{\mathcal{C}}^{0}$ together with the objects $X$ and $Y$. By virtue of Proposition 7.3.8.6, we can further assume that $\operatorname{\mathcal{C}}^{0}$ contains the object $X$. In this case, Proposition 8.5.1.8 implies that the functors $F$ and $U \circ F$ are left Kan extended from $\operatorname{\mathcal{C}}^{0}$, so that $F$ is $U$-left Kan extended from $\operatorname{\mathcal{C}}^{0}$ by virtue of Remark 7.3.3.20. $\square$