Definition [efr-002M]

Let \mathcal {M} be a monoidal category acting on \mathcal {C}. Then \mathsf {\mathbb Para}_\mathcal {M}(\mathcal {C}) is a pseudo double category where

1. The objects are the objects of \mathcal {C}
2. The vertical category is \mathcal {C}
3. The horizontal morphisms X \to Y are tuples (P \in \mathcal {M}, P \cdot X \to Y)
4. A 2-cell from (P,f): X \to Y to (P',f'): X' \to Y' with vertical boundary g: X \to X', h: Y \to Y' is a map P \to P' \in \mathcal {M} making the obvious diagram in \mathcal {C} commute.
5. The vertical composition of 2-cells in merely composition in \mathcal {M}. The horizontal composition is given by tensoring objects and morphisms.