Lax twisted arrow bicategory

Definition. Let $(C,\otimes ,I)$ be a bicategory. I call the lax-twisted arrow bicategory $\mathsf{Tw}(C)$ the monoidal bicategory where

• 0-cells are arrows $f:A\to B$ in $C$;
• 1-cells are lax-twisted squares;
• the monoidal unit is $\mathrm{id}:I\to I$;
• the monoidal tensor is the tensor of arrows;
• and 2-cells are cylindrical fillings.

Pseudomonoids in the lax-twisted arrow bicategory are Twisted arrow pseudomonoids.

Tags: arrow category, Monoids on the arrow category, bicategory.