Gluing a Chain of Noncommutative Couplings into Consistent Joint Laws

In the setting of Noncommutative Laws, Couplings and the Wasserstein Distance: Standing Notation, let $d\in\mathbb{N}$, let $(\lambda_{k})_{k\in\mathbb{N}}$ be a sequence in $\Sigma_{d}$, and for every $k\in\mathbb{N}$ let $\gamma_{k}\in\Pi(\lambda_{k},\lambda_{k+1})$ be a coupling. For $k\in\mathbb{N}$, with $kd$ the product of $k$ and $d$ and with the convention $(k-1)d+j=j$ when $k=1$, let

be the substitutions of the indicated tuples of variables. Then there is a sequence $(\Gamma_{k})_{k\in\mathbb{N}}$ with $\Gamma_{k}\in\Sigma_{kd}$ for every $k\in\mathbb{N}$ and the following properties.

1. (Start)¶ $\Gamma_{1}=\lambda_{1}$.

2. (Consistency)¶ $\Gamma_{k+1}\circ\beta_{k}=\Gamma_{k}$ for every $k\in\mathbb{N}$.

3. (Links)¶ $\Gamma_{k+1}\circ\delta_{k}=\gamma_{k}$ for every $k\in\mathbb{N}$.

4. (Blocks)¶ $\Gamma_{k}\circ\varepsilon_{k}=\lambda_{k}$ for every $k\in\mathbb{N}$.