Remark 7.3.1.11. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories, let $F_0: K \rightarrow \operatorname{\mathcal{D}}$ be a diagram, and let $\rho : \delta ' \rightarrow \delta $ be an isomorphism in the $\infty $-category $\operatorname{Fun}(K, \operatorname{\mathcal{C}})$. Then:
A natural transformation $\alpha : F \circ \delta \rightarrow F_0$ exhibits $F$ as a right Kan extension of $F_0$ along $\delta $ if and only if the composite natural transformation
\[ F \circ \delta ' \xrightarrow { \rho } F \circ \delta \xrightarrow {\alpha } F_0 \]exhibits $F$ as a right Kan extension of $F_0$ along $\delta '$ (note that this condition is independent of the composition chosen, by virtue of Remark 7.3.1.9).
A natural transformation $\beta : F_0 \rightarrow F \circ \delta '$ exhibits $F$ as a left Kan extension of $F_0$ along $\delta '$ if and only if the composite natural transformation
\[ F_0 \xrightarrow {\beta } F \circ \delta ' \xrightarrow {\rho } F \circ \delta \]exhibits $F$ as a left Kan extension of $F_0$ along $\delta $.
See Remark 7.1.1.8.