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Disintegrate $Q$ successively asFor each history $y_{<i}$, choose an optimal coupling of $P_i$ and $Q_i(\cdot \mid y_{<i})$. Sampling these couplings recursively produces a joint law of $(X,Y)$ with $Y\sim Q$. Its conditional $X_i$-marginal is always $P_i$ and is independent of the past, so $X\sim P_1\otimes \cdots \otimes P_N=P$. It is therefore a coupling $\pi \in \Pi (P,Q)$.

Write $c_i(y_{<i})$ for the conditional expected cost $w(X_i,Y_i)$ in the chosen coordinate coupling. The assumed one-coordinate transport-entropy inequality givesBy Jensen inequality and the chain rule for relative entropy,Taking the infimum over all couplings proves the tensorization of a transport-entropy inequality.