Thanks a lot, this is an excellent description on how to safely exchange the battery of my MacBook Pro 13" (early 2013, Model A1425)! However, I did skip all steps of removal of connectors other than the battery connector (see comments). Furthermore, I did not use solvent to remove the old batteries. I used bigger spudgers and some gentle force to lift the batteries off the housing. Afterwards, I used some acetone and a piece of cloth to remove the remaining film of glue. It took me about an hour (including cleaning). Everything worked absolutely perfect.

I also strongly recommend not to unmount the speaker plugs, just flip them out of the way, same as with the SSD drive. Makes it much less complicated and less risky. It worked perfect for me.

Ich empfehle ebenfalls, die Platinenstecker der Lautsprecher nicht zu lösen sondern die Lautsprecher einfach, so wie das SSD-Laufwerk, einfach aus dem Weg zu legen. Das macht die Sache weniger kompliziert und weniger risikoreich. Es hat bei mir super funktioniert!

I would recommend to use plastic tweezers in order to reduce risk of damage.

I also skipped this step as this data cable is not compromising the exchange of battery and less interaction with connectors is less risk of damage..

Diesen Schritt habe ich ebenfalls weggelassen, da das Datenkabel den Austausch der Batterie nicht behindert und weniger Manipulationen an Steckverbindern bedeutet ein geringeres Risiko für Beschädigungen