Let $\textbf{A}$ and $\textbf{B}$ denote a given sets. Suppose I have a conditional distribution $W_{Y|X}: \textbf{A} \rightarrow \textbf{B}$ where for each $a\in \textbf{A}$, $W_{Y|X}(\cdot | a)$ is a random variable on $\textbf{B}$.
Let $P$ and $Q$ be two other distribution of random variables on $A$ from which I want to generate 2 joint distribution: $P_{X,Y}=W_{Y|X}\cdot P$ and $Q_{X,Y}=W_{Y|X}\cdot Q$.
It this notation for the composition is acceptable? Is there other nicer way to or standard way to write this?