Alternatively my take is that while a function takes a thing of the domain and spits out another thing of the codomain, a conditional probability takes a thing

Alternatively my take is that while a function takes a thing of the domain and spits out another thing of the codomain, a conditional probability takes a thing of the domain and spits out a distribution on the things of the codomain. With P(c=C|d=D) with D=domain and C=codomain. With finite C and D that suffices. If the distributions are 1-hot you get a function.