Context
Let $\mathscr{G}$ be a $\sigma$-subalgebra of $\mathscr{F}$ and let $\mathbb{P}$ be a probability measure on a polish-space $X$.
Define the category $\mathfrak{C}$ of random-elements in $L^1(\mathbb{P};\mathscr{G})$ with arrows continuous maps of $X$ onto itself. Similarly, define the category $\mathfrak{D}$ of random-elements in $L^1(\mathbb{P};\mathscr{F})$ with arrows continuous maps of $X$ onto itself.
The conditional expectation $\mathbb{E}_{\mathbb{P}}\left[\cdot|\mathscr{G}\right]$ maps the objects of $\mathfrak{D}$ to those of $\mathfrak{C}$ and the precomposition map $\phi_{\star}$ defined by $$ \phi \rightarrow \mathbb{E}_{\mathbb{P}}\left[\phi(\cdot)|\mathscr{G}\right] . $$ Defines a map from $\mathfrak{D}$ to $\mathfrak{C}$. This makes conditional expectation into a functor
Question
Is there a way to canonical way of associating to every $\xi,\eta \in L^1(\mathbb{P};\mathscr{F})$, and every continuous map $\phi$ from $X$ to itself such that $$ \eta = \phi(\xi) $$ to a density $Z^{\phi} \in L^1(\mathbb{P};\mathscr{F})$ such that $$ \mathbb{E}_{\mathbb{P}}\left[\eta|\mathscr{G}\right]= \mathbb{E}_{\mathbb{P}}\left[Z^{\phi}\xi|\mathscr{G}\right] . $$