Remark 7.3.1.13 (Change of Target). 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, and let $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be a functor of $\infty $-categories. Then:
If $G$ is fully faithful and $G(\alpha ): (G \circ F) \circ \delta \rightarrow G \circ F_0$ exhibits $G \circ F$ as a right Kan extension of $G \circ F_0$ along $\delta $, then $\alpha $ exhibits $F$ as a right Kan extension of $F_0$ along $\delta $.
If $G$ is an equivalence of $\infty $-categories and $\alpha $ exhibits $F$ as a right Kan extension of $F_0$ along $\delta $, then $G(\alpha )$ exhibits $G \circ F$ as a right Kan extension of $G \circ F_0$ along $\delta $.
See Remark 7.1.1.10.