I followed the manuals on this Fairphone website and this. On the phone I got the message "Bootloader is already unlocked".
Then I start the script flash_fp5_factory.sh from the package FP5-TT4H-factory from Fairphone.
I got this message:

INFO: The procedure will start soon. Please wait...
INFO: The package type is "user-factory"
INFO: You are using a Linux distribution.
ERROR: The 'fastboot' tool is not available.
ERROR: Please install the tool using your package manager.
ERROR: Aborting now (press Enter to terminate).

In the shellscript "flash_fp5_factory.command" I see these lines:

Linux|GNU/Linux)
            echo "INFO: You are using a Linux distribution."
            # Find fastboot, abort if not available.
            FASTBOOT_BIN=$(which fastboot) || {
                echo -en "$RED"
                echo "ERROR: The 'fastboot' tool is not available."
                echo "ERROR: Please install the tool using your package manager."
                abort_now
            }

Does that mean that I also have to install the fastboot package on my PC?