Every society puts certain kinds of speech outside the window of respectability. Woke warriors on Twitter want to shrink the window and move it to the left, thereby branding some traditional ideas about identity as illegitimate. The right wants to use its political power to shrink the window and move it to the right, thereby silencing the woke warriors. Liberals want to leave the window as wide open as possible, and to keep it where it is.
That, in a nutshell, is what the battle over cancel culture is all about.