Let $(X\times Y,\mathcal{X}\otimes\mathcal{Y}, P)$ be a product probability space and $f_1, f_2$ the two projection maps. Let $Q:\mathcal{X}\times\Omega\rightarrow [0,1]$ be a conditional distribution of $f_1$ given $f_2$, i.e.~
$\omega\mapsto Q(A,.)$ is $Y$-measurable for any $A\in\mathcal{X}$;
$Q(.,\omega)$ is a probability distribution for any $\omega\in\Omega$;
$P(A\cap B)=\int_BQ(A,\omega)dP$ for any $A\in\mathcal{X}$ and $B\in\mathcal{Y}$.
My question is whether there is a canonical extension of this $Q$ to a regular conditional distribution of $P$ given $f_2$ (which is a function $Q': \mathcal{X}\otimes\mathcal{Y}\times \Omega\rightarrow [0,1]$). Right now I know how to define $Q'(A\times B,\omega)=Q(A,\omega)1_B(\omega)$. So the question could also be framed as one about extending a regular conditional distribution defined on $\mathcal{C}\times\Omega$ to $\sigma C\times\Omega$. Thanks in advance for any help!
(I have tried to extend $Q(.,\omega)$ pointwise by the Caratheodory extension theorem, but I don't know how to prove that this extension preserves measurability...)