Unitors in Morton's definition of a double bicategory

Question

I am confused by the definition of a double bicategory by Morton in (Definition 3.1.1. in https://arxiv.org/abs/math/0611930), but I need it, so I want to make sure I understand it correctly before I proceed. In paticular, Morton does not give an identity morphism $Obj \rightarrow Mor$, but instead two unitors $l, r: Obj \rightarrow Mor$. But if this is just spelling out the definition of a bicategory internal to $Bicat$, then shouldn't the unitors go from $Mor$ to $2-Mor$? And where is the identity morphism?