I have two sampling surjective techniques $\varphi_1,\varphi_2:[0,1)\to E$ mapping a random number $u\in[0,1)$ to a sample in a measurable space $(E,\mathcal E)$.
Say $u\in[0,1)$ and $x:=\varphi_1(u)$. Is there any chance to compute the random number $v\in[0,1)$ which would have produced the same sample $x$ under $\varphi_2$, i.e. $x=\varphi_2(v)$?
Clearly, if (for example) $\varphi_1$ is not injective, $\varphi_1$ might be constant on an interval and hence we're not able to identify a unique random number producing $x$. However, maybe we can somehow commit to a single value.