Remark 2.2.1.6. Let $\operatorname{\mathcal{C}}$ be a $2$-category. Then $\operatorname{\mathcal{C}}$ can be obtained from an ordinary category (via the construction of Example 2.2.0.6) if and only if every $2$-morphism in $\operatorname{\mathcal{C}}$ is an identity $2$-morphism (note that a $2$-category with this property is automatically strict, by virtue of Example 2.2.1.4).
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$