Corollary 7.3.3.22. Suppose we are given a pullback diagram of $\infty $-categories
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{D}}' \ar [d]^{U'} \ar [r]^-{G} & \operatorname{\mathcal{D}}\ar [d]^{U} \\ \operatorname{\mathcal{E}}' \ar [r] & \operatorname{\mathcal{E}}, } \]
where the vertical maps are inner fibrations. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}'$ be a functor of $\infty $-categories and let $\operatorname{\mathcal{C}}^{0} \subseteq \operatorname{\mathcal{C}}$ be a full subcategory. If $G \circ F$ is $U$-left Kan extended from $\operatorname{\mathcal{C}}^{0}$, then $F$ is $U'$-left Kan extended from $\operatorname{\mathcal{C}}^{0}$. The converse holds if $U$ is a cartesian fibration.