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Does the plastic cover adhered to the battery inside the Macbook pro 13in retina early 2013 have to be replaced after being removed? I Ive tried just sticking the original back on but it doesn’t stick very well. Can I just remove it?

If you are referencing the cover the came with the new battery, it is just there to prevent the battery from sticking to things. It can be discarded after the battery is installed.