# Kerodon

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Remark 7.1.4.9. Suppose we are given a commutative diagram of $\infty$-categories

$\xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}\ar [d]^{U} \ar [r]^-{F} & \operatorname{\mathcal{C}}' \ar [d]^{U'} \\ \operatorname{\mathcal{D}}\ar [r] & \operatorname{\mathcal{D}}', }$

where the horizontal maps are equivalences of $\infty$-categories. Then an object $X \in \operatorname{\mathcal{C}}$ is $U$-initial if and only if $F(X) \in \operatorname{\mathcal{C}}'$ is $U'$-initial, and $U$-final if and only if $F(X)$ is $U'$-final.