If $z=(x, v) \sim \pi(dx)\eta(dv)$, find $\eta(dv)$ such that $v'\sim\varpi$ where $z'=\psi(z)$ and $\psi$ invertible

Question

I have a variable $z = (x, v)$ distributed according to $\lambda(dz) = \pi(dx)\eta(dv)$, where $\pi$ is fixed and known, but $\eta$ is unknown.

I also have an invertible transformation $\psi$ and I compute $z' = \psi(z)$, with $z' = (x', v')$.

Can I choose $\eta(dv)$ so that $v' \sim \varpi$? I don't care what the distribution of $x' \mid v'$ is, could be anything.

Optimal Transport Formulation

If $z\sim \pi\otimes \eta$ for $\pi$ known and $\eta$ unknown, how can I choose $\eta$ so that $\Pi_v(\pi\otimes \eta) = \varpi$? Here $\Pi_v$ is the projection operation, that marginalizes basically.