Remark 7.3.1.9. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories and let $\delta : K \rightarrow \operatorname{\mathcal{C}}$ and $F_0: K \rightarrow \operatorname{\mathcal{D}}$ be diagrams. Then:
The condition that a natural transformation $\alpha : F \circ \delta \rightarrow F_0$ exhibits $F$ as a right Kan extension of $F_0$ along $\delta $ depends only on the homotopy class $[\alpha ]$ (as a morphism in the $\infty $-category $\operatorname{Fun}( K, \operatorname{\mathcal{D}})$).
The condition that a natural transformation $\beta : F_0 \rightarrow F \circ \delta $ exhibits $F$ as a left Kan extension of $F_0$ along $\delta $ depends only on the homotopy class $[\beta ]$ (as a morphism in the $\infty $-category $\operatorname{Fun}( K, \operatorname{\mathcal{D}})$).
See Remark 7.1.1.7.