  Several well-studied probability monads have been expressed as codensity
monads over small categories of stochastic maps, giving a limit description of
spaces of probability measures. In this paper we show how properties of
probability monads such as commutativity and affineness can arise from their
codensity presentation. First we show that their codensity presentation is
closely related to another characterisation of probability monads as terminal
endofunctors admitting certain maps into the Giry monad, which allows us to
generalise a result by Van Breugel on the Kantorovich monad. We then provide
sufficient conditions for a codensity monad to lift to $\bf{MonCat}$, and give
a characterisation of exactly pointwise monoidal codensity monads; codensity
monads that satisfy a strengthening of these conditions. We show that the Radon
monad is exactly pointwise monoidal, and hence give a description of the tensor
product of free algebras of the Radon monad in terms of Day convolution.
Finally we show that the Giry monad is not exactly pointwise monoidal due to
the existence of probability bimeasures that do not extend to measures,
although its restriction to standard Borel spaces is. We introduce the notion
of a $*$-monad and its Kleisli monoidal op-multicategory to describe the
categorical structure that organises the spaces of probability polymeasures on
measurable spaces.
