We establish a Brenier theorem for iterated Wasserstein spaces. Specifically, for a separable Hilbert space $H$ and $N\ge 1$, we construct a full-support probability $\Lambda\in \mathcal{P}_2^{N}(H)= \mathcal{P}_2(\ldots \mathcal{P}_2(H)\ldots)$ that is transport regular: for all $P,Q\in \mathcal{P}_2^{N}(H)$ with $P\ll \Lambda$, the $\mathcal{W}_2^2$-optimal transport from $P$ to $Q$ is unique and of Monge type. In the first non-classical case $N=2$ we show that optimal transports are given as the push-forward by the $\mathcal{W}_2$-gradient (or Lions' derivative) of an $L$-convex function. To establish the result for general $N$ we develop new adapted notions of Lions' lift, $L$-convexity and Lions' derivative. A key idea is a new identification between optimal-transport $c$-conjugation (with $c$ given by maximal covariance) and classical convex conjugation on the lift.

A primary motivation comes from the adapted Wasserstein distance $\mathcal{AW}_2$: our results yield a first Brenier theorem for $\mathcal{AW}_2$ and characterize $\mathcal{AW}_2^2$-optimal couplings through convex functionals on the space of $L_2$-processes.