Is defining sampling as a functor from the category of stochastic relations to a sub category of sets where each object is the product between a set and a (finite or infinite) set of independent samples, valid?
The sample functor would send each object in SRel, a random variable in some domain, to a member of the product of a member of that domain and a sample.
And for morphisms, the functor would send each morphism to a function from the co-domain and a sample to the codomain and another sample $ F(I \rightarrow O) = I \times S \rightarrow O \times S$
This seems to be able to preserve the identity and composition, but is this the category theoretic way of going about things?