So there is already a name for a bijection $f$ between a topological space $(X, \tau_{1})$ and a topological space $(Y, \tau_{2})$ such that $f$ is continuous-$(\tau_{1}, \tau_{2})$ and $f^{-1}$ is continuous-$(\tau_{2}, \tau_{1})$. I wonder, if we replace "topological" with "measurable", $\tau_{1}, \tau_{2}$ with sigma algebras, and "continuous" with "measurable", then how do people call $f$?
Such a map, if I am not mistaken, is used in probability theory for defining the concept of a Borel space. (If a measurable space is identified as the Borel space $\mathbb{R}$ under such a map, then it is called a Borel space.) But of interest now is the "canonical" name of the map itself.