It is known that a measurable bijection $f \colon [0,1] \to [0,1]$ has a measurable inverse. (Here, all measurability is simply with respect to the Borel $\sigma$-algebra of $[0,1]$.)

Now fix an arbitrary measurable space $(\Omega,\mathcal{F})$, and let $(f_\omega)_{\omega \in \Omega}$ be a family of bijections $f_\omega \colon [0,1] \to [0,1]$ such that the map $(\omega,x) \mapsto f_\omega(x)$ is $(\mathcal{F}\otimes\mathcal{B}([0,1]),\mathcal{B}([0,1]))$-measurable. Is it necessarily the case that the map $(\omega,x) \mapsto f_\omega^{-1}(x)$ is $(\mathcal{F}\otimes\mathcal{B}([0,1]),\mathcal{B}([0,1]))$-measurable?