Can I update add-ons on my own or do I have to wait for new MazzleUI versions?
You can update most add-ons on your own with a few caveats.
First, it's possible the Mazzifier will no longer configure the settings properly. If the author decides to change the format of the settings, Mazzlefizz must manually make the same changes in the Mazzifier. If the Mazzifier creates settings that don't make sense to the newer version of the add-on, bad things can happen. If you do update an add-on, I recommend that you do not re-mazzify that add-on until a new version of the Mazzifier comes out.
Second, you need to remember that Mazzlefizz has modified many add-ons to add extra features, to improve performance or to just integrate the add-on into the UI. In some cases, upgrading won't break anything but you will lose those features that were added. In other cases, things may break and you'll get errors. Unless you have a good reason, I'd suggest waiting for the next update of MazzleUI. That should be the easiest way to make sure you always have the latest and greatest versions of add-ons at your disposal.
Here is a list of add-ons that have modifications or overrides, to help you decide how problematic it may be to upgrade a particular add-on on your own.
The addons that I'd most want to update are things like PeriodicTable and Mendeleev, to keep up to date with new game items as they're added and classified. (I'm kind of assuming that the writers of such mods are constantly catching up with new item data.)
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Author of GemSay, which is MazzleUI compatible!