What category of categories have faithful functors into Markov categories? We know that various Markov categories have deterministic morphisms.  This suggests that they support faithful functors into them from categories of categories.  One interesting Markov category is the Kleisli Category of the Distribution Monad, KlDM.  I need to know if KlDM supports a faithful functor into it from one or both of the following categories:

*

*The category of small categories

*The category of finitely presented categories

*The category of concrete categories

 A: Following N. Virgo's point (but even more simply), any concrete category $\mathcal C$ admits (by definition) a faithful functor $F: \mathcal C \to Set$. The free functor $G$ from $Set$ to the Kleisli category of the distribution monad is faithful as well. So $GF: \mathcal C \to Kl(Dist)$ is a faithful functor. Conversely, if a category $\mathcal C$ admits a faithful functor $H: \mathcal C \to Kl(Dist)$, then since the forgetful functor $U: Kl(Dist) \to Set$ is also faithful, it follows that $UH$ is faithful, i.e. $\mathcal C$ is concrete. That is, $\mathcal C$ admits a faithful functor to $Kl(Dist)$ iff $\mathcal C$ is concrete.
For example, the category of small categories is concrete (the functor $Mor : Cat \to Set$ which carries a category to its set of morphisms is faithful). Likewise, so is the category of small finitely-presented categories. So these categories are concrete / admit faithful functors to $Kl(Dist)$.
The category of concrete categories is not locally small for any reasonable notion of morphism I can think of, so it is not concrete / it does not admit a faithful functor to $Kl(Dist)$.
