Remark 7.3.1.13. Suppose we are given a diagram
as in Definition 7.3.1.2. Let $\rho : F' \rightarrow F$ be a morphism in the $\infty $-category $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$. Then any two of the following conditions imply the third:
The natural transformation $\alpha $ exhibits $F$ as a right Kan extension of $F_0$ along $\delta $.
The composite natural transformation
\[ \delta \circ F' \xrightarrow { \rho } \delta \circ F \xrightarrow { \alpha } F_0 \]exhibits $F'$ as a right Kan extension of $F_0$ along $\delta $ (note that this condition does not depend on the composition chosen, by virtue of Remark 7.3.1.10).
The morphism $\rho $ is an isomorphism in the $\infty $-category $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$.
This follows by combining Remark 7.1.1.9 with Theorem 4.4.4.4.