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I don't think the narrator adapting to the name change would be a problem per se. However, what needs to be clear to the reader is which character you are referring to. So, suppose that we have a ...
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#3: Attribution notice added
Source: https://writers.stackexchange.com/a/39754 License name: CC BY-SA 3.0 License URL: https://creativecommons.org/licenses/by-sa/3.0/
#2: Initial revision
I don't think the narrator adapting to the name change would be a problem _per se_. However, **what needs to be clear to the reader is which character you are referring to**. So, suppose that we have a scene where Jane and Dolores meet and agree to switch names. As a reader, if you then tell me that «Jane left the building and returned home», I am left wondering _which Jane_ did it. It would be much better that you clarified that _the new Jane left the building_. Or even _the old Jane, that was now called Dolores, left the building_... This way, we disambiguate it for now, but will need to clarify who is when we the narrator only talks about _Jane_. I would then eg. start the next chapter with «Jane woke up, looked at herself in the mirror and said "Hello Dolores" [or "Hello Jane"]. She was still trying to get used to that name…» which is quite similar to the idea you had. Then, when the two characters meet another time, you should disambiguate again. Just think on how confusing it may be having the narrator refer to the characters in one way, they themselves in another or even differing between them (eg. the old Dolores may think in herself as Jane now, but also think in the new Dolores as Jane, as that's how she met her). Even if there was only one valid interpretation, the reader shouldn't need to stop reading to test possibilities and figure out who-is-who (not to mention that these situations are likely to confuse the writer, too!).