Suppose that $X,Y,Z \in L^2(\Omega,\mathcal{F},\mathbb{P};\mathbb{R}^d)$, are $d$-dimensional random-vectors and there exists functions $f,g\in L^2(\mathbb{R}^d,\mathcal{B}(\mathbb{R}^d),Law(X);\mathbb{R}^d)$ satisfying $$ f(X)=Y \qquad g(X)=Z. $$
If $f$ is invertible, then the function $h\triangleq g\circ f^{-1}$ implies that $$ Z=h(Y). $$
However, if $f$ is not invertible, when does there exists a function $h$ satisfying $ h\circ f = g $ except possibly on a set of $Law(X)$-measure $0$? Hence $$ h(Y)=Z . $$