What is the connection between Goodbye to Berlin and Christopher and His Kind, and to what extent do the two books share a narrator?

Goodbye to Berlin is a novel about Isherwood's time in Berlin before World War Two. It is semi-autobiographical, and many of the stories he tells about Berlin are true, but it is also fictionalized. Isherwood himself describes himself as "a camera, recording, not thinking," and this is reflected in the fact that most of the stories he tells do not reveal very much about himself at all. The famous "Sally Bowles" character in this novel (more a collection of short stories, really, connected by a thread) is fictionalized and given a new name. Isherwood also left out a good deal about the main reason he went to Berlin in the first place.
In a way, this was what drove him to write Christopher and His Kind, several decades later. Christopher and His Kind is pure autobiography and is not fictionalized. It tells some of the same stories as Goodbye to Berlin but acknowledges that Isherwood and Auden mainly went to Berlin because it was known for its thriving gay underworld (not even very much hidden, because of the permissiveness of Berlin in the Weimar years). The title of the book refers to "his kind," meaning "other gay men," and this—gay experience in pre-War Berlin—is very much the focus of the book. It fills in all the gaps that seem to exist in Goodbye to Berlin about why the narrator was there and sort of turns the camera on him for once.