Remark 2.2.8.23. Let $\operatorname{\mathcal{C}}$ be a $2$-category and let $f,g: X \rightarrow Y$ be $1$-morphisms in $\operatorname{\mathcal{C}}$ having the same source and target. If $f$ and $g$ are isomorphic (as objects of the category $\underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y)$), then $f$ is an isomorphism if and only if $g$ is an isomorphism.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$