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I replaced my iPhone 6s screen transferring over the original home button, original camera and original sensors. Everything is OEM except for the screen which is 3rd party. The touch ID works after replacement, but I want to know if I can update safely through iTunes on the PC without errors (and if I can, should I update). I want my iPhone to be able to use touch ID to unlock the device after the update to 13.4.
Yes you can update your device to the latest firmware without any issues. Apple has attempted to brick devices after repair in the past but have been served class action lawsuits as a result. They tried to do that in late 2017 or early 2018 with the 8 and X by serializing the display. If the device didn’t detect the correct serial number, the phones would get bricked. But like I said, you should be fine.