Definition Markov Promonad [efr-UVD1]

Let \mathcal {C} be a Markov category. Recall that a promonad may be defined as an identity-on-objects functor \mathcal {C} \to \mathcal {C}_s. If \mathcal {C}_s admits a symmetric monoidal structure which makes this a strict monoidal functor, and which is furthermore semicartesian, observe that the copy maps in \mathcal {C} push forward to copy maps in \mathcal {C}_s, so that \mathcal {C}_s acquires a unique Markov structure where this is a Markov functor. In this case we call this promonad a Markov promonad.