# Kerodon

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Remark 7.3.1.12. Suppose we are given a diagram

$\xymatrix@R =50pt@C=50pt{ & \operatorname{\mathcal{C}}\ar [dr]^{F} \ar@ {=>}[]+<0pt,-25pt>;+<0pt,-50pt>^-{\alpha } & \\ K \ar [ur]^{\delta } \ar [rr]_{F_0} & & \operatorname{\mathcal{D}}}$

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 7.3.1.9).

• 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.