@John - I was looking at your old [Network Theory III](http://math.ucr.edu/home/baez/networks_oxford/networks_entropy.pdf) notes and I saw a familiar pattern:

> Given finite sets \$$X\$$ and \$$Y\$$ , a stochastic map \$$f : X \rightsquigarrow Y\$$ assigns a
> real number \$$f\_{yx}\$$ to each pair \$$x \in X\$$, \$$y \in Y\$$ in such a way that for
> any \$$x\$$, the numbers \$$f\_{yx}\$$ form a probability distribution on Y .
> We call \$$f\_{yx}\$$ the probability of \$$y\$$ given \$$x\$$.
>
> So, we demand:
>
> - \$$f\_{yx} \geq 0\$$ for all \$$x ∈ X\$$ \$$y ∈ Y\$$
> - \$$\displaystyle{\sum\_{y\in Y} f\_{yx} = 1} \$$ for all \$$x \in X\$$
>
> We can compose stochastic maps \$$f : X \rightsquigarrow Y\$$ and \$$g : Y \rightsquigarrow Z\$$ by
matrix multiplication:
>
> $(g \circ f )_{zx} = \sum_{y\in Y} g_{zy} f_{yz}$
>
> and get a stochastic map \$$g ◦ f : X \rightsquigarrow Z\$$.
>
> We let \$$\mathtt{FinStoch}\$$ be the category with
>
> - finite sets as objects,
> - stochastic maps \$$f : X \rightsquigarrow Y\$$ as morphisms.

It is perhaps well beyond the scope of this course, but is \$$\mathtt{FinStoch}\$$ a \$$\mathcal{V}\$$-enriched profunctor category? It doesn't seem like it is because \$$\sum\$$ is not the same as \$$\bigvee\$$. Is it some kind of related profunctor category?