They introduce a new correlation. For $\pi\in \Pi(\mu,\nu)$ the set of coupling of two probability measures $\mu$ and $\nu$ on a Polish space $(X,d)$. The author introduces a plugin estimator.

Question: How to understand $\hat{\pi}$?

For the estimator of the denominator, I use the same notation as the answer.

\begin{equation*} D(\hat{\pi})=\sum_{y,z}d(y,z)\hat{\pi}_2(\{y\})\hat{\pi}_2(\{z\}) =\sum_{y,z}d(y,z)\frac{1}{N^2}\sum 1[\phi(Y_n)\in\{y\}]\sum 1[\phi(Y_m)\in\{z\}] \end{equation*} Expand the product of these two summation, $$ =\frac{1}{N^2}\sum_{y,z}d(y,z)(\sum_{i=1}^N 1[\phi(Y_i)\in\{y\}]1[\phi(Y_i)\in\{z\}]+\sum_{i\neq j}1[\phi(Y_i)\in\{y\}]1[\phi(Y_j)\in\{z\}] ) $$