Remark 2.2.4.2. In the situation of Definition 2.2.4.1, conditions $(3)$ and $(4)$ are automatically satisfied if the $2$-categories $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ are strict.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$