Warning 2.2.5.11. The construction of Remark 2.2.5.10 requires that the identity and composition constraints of $F$ are invertible, and therefore does not extend to lax functors between $2$-categories. In general, one cannot identify lax functors from $\operatorname{\mathcal{C}}$ to $\operatorname{\mathcal{D}}$ with lax functors from $\operatorname{\mathcal{C}}^{\operatorname{c}}$ to $\operatorname{\mathcal{D}}^{\operatorname{c}}$: the definition of lax functor is asymmetrical with respect to vertical composition.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$