Theorem [efr-003R]

Let \mathbb {M} be a monoidal strict double category, which acts on a strict double category \mathbb {C}. Then for each n \in \Delta ^\mathrm {op}, we have an actegory \mathbb {M}[n] acting on \mathbb {C}[n]. This yields a bisimplicial category [n,m] \mapsto \mathsf {\mathbb Para}_{\mathbb {M}[m]}(\mathbb {C}[m])[n]. This is an iwss triple category

In particular, let \mathbb {M} \to \mathbb {C} be a functor of symmetric monoidal strict double categories. Then the canonical action of \mathbb {M} on \mathbb {C} gives a double category as above.