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Log-in to your server account using FileZilla, browse to the PrestaShop folder, and find the folder that are marked by the installer as in need of a permission change. Changing file/folder permission on a Unix/Linux system is called a "CHMOD", after the command of the same name – you can find an explanation of file permissions here). Giving them "write permission" is called "doing a CHMOD 775755" or "a CHMOD 777775, depending on the host – although the last one is not recommended for anything more than a one-time need.
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