Example [efr-MN7C]

Let \mathcal {C} be a monoidal category, and let S: \mathcal {C} \to \mathsf {Set} be any functor. Recall that \int S is the category whose objects are pairs (X \in \mathcal {C}, s \in S(X)) and whose morphisms (X,s) \to (Y,s') are f: X \to Y so that S(f)(s) = s'. If S is lax monoidal (for (\mathsf {Set},\times ),) \int S acquires a lax monoidal structure making the forgetful functor \int S \to \mathcal {C} strong (even strict) monoidal.

With the action induced by this functor, the bicategory \mathsf {Para}_{\int S}(\mathcal {C}) has morphisms given by maps M \otimes X \to Y, m \in S(M), and maps given by reparametrizations which preserve the decoration m.

For example, let \mathcal {C} be a Markov category and let S(X) = \mathcal {C}(I,X). Then morphisms are parametrized maps equipped with a measure on the parameter space.