This question is mostly for the purpose of learning how to reason in category theory.
Consider the category of finite stochastic maps. I imagine this is something fairly standard, but I've given a definition below in case it's not. I'll call this category FinStochMap.
Products, coproducts, and the initial and final objects in FinStochMap are the same as in FinSet. In FinStochMap, a morphism from the terminal object to an object $A$ can be seen as a probability distribution over $A$. To be concrete, consider a morphism in FinStochMap, $f:\{*\}\to A$. This gives us a set of numbers $p(a|*)$ for every $a\in A$, such that $\sum_{a\in A} p(a|*)=1$, so we can interpret $p(a|*)$ as the probability of $a$.
In probability theory, an important kind of probability distribution is the uniform distribution. Among other things, this is the only distribution on a finite set that is invariant to permutations of its elements.
My question is, is there a category-theoretic way to pick out the uniform probability distributions in FinStochMap? Are they the morphisms from the terminal object that obey a certain universal property, and if so, what is it?
Definition of FinStochMap. By FinStochMap I mean the category where the objects are finite sets, and the morphisms are conditional probability distributions (or Markov kernels if you prefer), i.e. a morphism $f:A\to B$ is a function $p_f:A\times B \to [0,1]$, with elements written $p_f(b|a)$, such that $\sum_{b\in B} p(b|a)=1$ for all $a\in A$. The composition rule is that for $f:A\to B$ and $g:B\to C$, the composition $f;g:A\to C$ is given by $$ p_{f;g}(c|a) = \sum_{b\in B} p_f(b|a)p_g(c|b). $$ An identity morphism $I_A:A\to A$ is given by $p_{I_A}(a|a') = 1$ if $a=a'$, and 0 otherwise.
Related question by me: expectations in FinStochMap.
This is a self-answer based on Oscar Cunningham's comment.
The automorphisms in FinStochMap are exactly the permutations of a set. That is, if $f:A\to A$ is an isomorphism then each $p_f(a'|a)$ must either be 0 or 1, with exactly one 1 on each row and each column, if we view it as a matrix. Then we can just define the uniform distributions as those distributions that are automorphism invariant.
That is, a morphism $u_A:\{*\}\to A$ is a uniform distribution if, for every automorphism $f:A\to A$ we have $u_A f = u_A$.
Equivalently, we could say that $u_A:\{*\}\to A$ is a uniform distribution if for every pair of isomorphisms $f, g:A\to B$ we have $u_A f = u_A g$.
In his comment Oscar Cunningham worried that this would be 'boring' (i.e. unsatisfying), but I think it's fine - it's a perfectly good category-theoretic definition, because it only relies on the 'external' features of the morphisms, not their internal structure, and it allows us to see that (for example) FinSet does not have the analogous feature. (In FinSet the morphisms from the terminal object correspond to set membership, and only the terminal object has an automorphism-invariant member.)