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My harddisk on my iMac died, so I replaced it with a new compatible one. But now I have no idea what to do software-wise. Do I need an adapter to upload a OS-version to it first, before installation, or do I upload the software after the installation of the harddisk?

Once you get the hardware side of things working you’ll need to create a bootable OS installer drive. I would recommend staying with Sierra on your system, here’s what you’ll need to do from a second Mac system How to create a bootable macOS Sierra installer drive. If you have a good internet connection you could try the OS install from the internet How to start up your Mac in Internet Recovery Mode. As you have a fresh drive you don’t have the recovery partition to work off of.