I have recently been guided towards a model of probabilistic functions. A probabilistic function which models functions of the form $f : X \rightarrow Y$, is a map from $X$ into probability distributions over $Y$. It has been suggested that these maps are actually morphisms in a category $Prob$. This means that there is a standard way of composing them to mimic function composition. Furthermore, there should be functors from any concrete category $C$ into $Prob$, and there is a faithful functor that maps morphisms in $C$ into morphisms where the codomain distributions are zero everywhere except at one set element. I understand that this is related to the Giry monad, where we send a measurable set to the set of probability measures on that set. I am told this is related to Lawvere theories.
Can someone verify this, perhaps expound on it, and give me a reference? In this discipline, when/why do we talk about the category $Prob$ vs the probability monad?