Let $p$ be a probability (or Markov) kernel with source $(X,\mathcal{A})$ and target $(Y,\mathcal{B})$, and $q$ a probability kernel with source $(Y,\mathcal{B})$ and target $(Z,\mathcal{C})$.
We can prove (in a similar manner as in the existence of the product measure, using Carathédory's extension theorem for instance) that the function $$r(x,B \times C) = \int_B q(y,C) p(x,dy)$$ can be uniquely extended to a probability measure on the product $\sigma$-algebra $\mathcal{B} \otimes \mathcal{C}$ for every $x \in X$. I also denote $r$ this extension.
How can we prove that $r$ is a probability kernel with source $(X,\mathcal{A})$ and target $(Y \times Z,\mathcal{B} \otimes \mathcal{C})$, namely $x \mapsto r(x,T)$ is $\mathcal{A}$-measurable for every $T \in \mathcal{B} \otimes \mathcal{C}$?