For Arch Linux, edit /etc/pacman.conf and add the following (note that the order of repositories in pacman.conf is important, since pacman always downloads the first found package):
[home_misotolar_linux-cachyos_Arch]
Server = https://download.opensuse.org/repositories/home:/misotolar:/linux-cachyos/Arch/$arch
Then run the following as root
key=$(curl -fsSL https://download.opensuse.org/repositories/home:misotolar:linux-cachyos/Arch/$(uname -m)/home_misotolar_linux-cachyos_Arch.key)
fingerprint=$(gpg --quiet --with-colons --import-options show-only --import --fingerprint <<< "${key}" | awk -F: '$1 == "fpr" { print $10 }')
pacman-key --init
pacman-key --add - <<< "${key}"
pacman-key --lsign-key "${fingerprint}"
pacman -Sy home_misotolar_linux-cachyos_Arch/linux-cachyos