Let F: \mathcal {C} \to \mathcal {D} be a functor between Markov categories.
If F is a Markov functor, it preserves deterministic maps. The oplax monoidal structure on F is necessarily given by the unique one induced on F_\mathrm {det} : \mathcal {C}_\mathrm {det} \to \mathcal {D}_\mathrm {det}.
Moreover it preserves independent pairings, in the sense that
commutes, where \Delta is the oplax monoidal structure.
Given a functor F': \mathcal {C} \to \mathcal {D} which preserves deterministic maps and independent pairings in this sense, the comonoid structure on F'_\mathrm {det} makes F into an oplax Markov functor.
A Markov functor is strong if and only if F_\mathrm {det} preserves products.
