Plans on the Product of a Euclidean Space with Itself

In the setting of Probability Measures on Euclidean Space and Random Vectors: Standing Notation, let $d$ be a natural number.

(i) A plan in dimension $d$ is an element of the set $\mathcal{P}_{2}(\mathbb{R}^{d+d})$ of probability measures on $\mathbb{R}^{d+d}$ with finite second moment, that definition being read with the natural number $d+d$ in place of the dimension written $d$ there.