If $X$ is a continuous random variable with known distribution, and $Y_1= f_1(X)$, $Y_2= f_2(X)$ where $f_1$ and $f_2$ are strictly increasing functions and distribution of $Y_1$ and $Y_2$ is the same, does this imply the functions $f_1 = f_2$?
In other words, there is a unique strictly increasing function that transfers a source distribution to a destination distribution.