Maybe there is an easier way, but it seems that the only way to add or remove a category is to notice the little pencil, go in and make changes and then make sure to press save?
It’d be nice if there was something like a “Follow/Unfollow” button available when viewing a category.
This is not the only, but the most simple way to enter categories to the sidebar.
If you want to follow up a certain category permanently you can also do that by setting it to “watch” or “track” or “watch first post” by clicking the “bell” button on the right side of the header (just beneath your profile image) and choose what you like. The category will not appear in the sidebar then but you will get notification whenever new topics or posts are available and they will be entered into your “new” and “unread” list. Maybe you are already aware of that - just wanted to point this option.
I personally don’t see much utility in an extra button on the category page. For me, the pencil icons on the sidebar are the most obvious and intuitive place to edit the links.
It would likely also get confusing next to the other options that already exist.