Let $(P_i,Q_i)$ be couples of distributions, $i = 1,...n$. Suppose they share the same Markov kernel K, that is, $\forall i\in\{1,...n\}$ : $$ Q_i(dy) = \int K(x,dy)P_i(dx) $$
Now let $(X_i,Y_i)$ be a coupling with marginals $(P_i,Q_i)$ and conditional probability $K$.
Is there a function $h$ independent of i such the following equality holds in distribution for all i: $$ Y_i = h(X_i) $$