Let $\mathcal{A},\mathcal{B}\,$ be $\sigma$-algebras on sets $X,Y$, respectively. A mapping $h\colon\mathcal{A}\to\mathcal{B}\,$ is called a $\sigma$-complete homomorphism if $h(\emptyset)=\emptyset$, $h(X\setminus A)=Y\setminus h(A)$ for all $A\in\mathcal{A}$, and $h\big(\bigcup A_n\big)=\bigcup h(A_n)$ for all sequences $\{A_n\}_{n\in\omega}$ in $\mathcal{A}$.
I am looking for an example of such $\mathcal{A}$, $\mathcal{B}$ and $h$ for which there is no function $f\colon Y\to X$ satisfying $h(A)=f^{-1}[A]$ for all $A\in\mathcal{A}$.
If $\kappa$ is a measurable cardinal then one can take a $\sigma$-complete non-principal ultrafilter $u$ on $\mathcal{P}(\kappa)$ and define $h\colon\mathcal{P}(\kappa)\to\mathcal{P}(\{\emptyset\})$ by taking $h(A)=\{\emptyset\}$ if and only if $A\in u$.
Question: Can one find an example of such $\mathcal{A}$, $\mathcal{B}$ and $h$ in ZFC?
Let $X$ be an uncountable set and let $\mathcal{A}$ be the algebra of countable or cocountable subsets of $X$. Then there is a $\sigma$-complete homomorphism $h:\mathcal{A}\to\{0,1\}$ given by $h(A)=1$ if $A$ is cocountable and $h(A)=0$ if $A$ is countable. Identifying $\{0,1\}$ with the power set of a singleton set $Y$, $h$ is not induced by any function $Y\to X$.