Proposition [efr-002V]

Let \mathcal {C} be enriched in \mathcal {M}. Then this corresponds to a bicategory over B\mathcal {M}. Denote it \mathsf {Para}^\mathcal {M}(\mathcal {C}). The objects are the same as \mathcal {C}, and a morphism consists of a tuple M \in \mathcal {M}, M \to \mathcal {C}(X,Y) \in \mathcal {M}.

Let \mathcal {C} carry an action of \mathcal {M}. This corresponds to a bicategory of B\mathcal {M}. Denote it \mathsf {Para}_\mathcal {M}(\mathcal {C}). The objects are the same as \mathcal {C}, and a morphism X \to Y consists of a tuple M \in \mathcal {M}, X \to M \cdot Y.