Para construction for double categories

We introduce Para and coPara double categories for double categories. They rely on a horizontal action $\text{crta}\text{ot}$ of a horizontally monoidal double category $\text{Mm}$ on a double category $\text{Dd}$. We prove a series of properties, most importantly, we characterize monoidality of ${\text{coPara}}_{\text{Mm}}(\text{Dd})$ in the way that it extends monoidality of $\text{Dd}$ as: lax monoidality of the action $\text{crta}\text{ot}$; bistrong commutativity of $\text{crta}\text{ot}$; and purely central premonoidality of ${\text{coPara}}_{\text{Mm}}(\text{Dd})$.