Remark 7.3.3.15. Suppose we are given a commutative diagram of $\infty $-categories
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{D}}\ar [r]^-{G} \ar [d]^{U} & \operatorname{\mathcal{D}}' \ar [d]^{U'} \\ \operatorname{\mathcal{E}}\ar [r] & \operatorname{\mathcal{E}}', } \]
where the horizontal functors are equivalence of $\infty $-categories. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor and let $\operatorname{\mathcal{C}}^{0} \subseteq \operatorname{\mathcal{C}}$ be a full subcategory. Then $F$ is $U$-left Kan extended from $\operatorname{\mathcal{C}}^{0}$ if and only if $G \circ F$ is $U'$-left Kan extended from $\operatorname{\mathcal{C}}^{0}$ (see Remark 7.1.6.6). Similarly, $F$ is $U$-right Kan extended from $\operatorname{\mathcal{C}}^{0}$ if and only if $G \circ F$ is $U'$-right Kan extended from $\operatorname{\mathcal{C}}^{0}$.