what is ‘fan edits’ or ‘edits’ in chinese? Like you know those mini edited clips of a celebrity/show..with music and stuff. Surely there’s a Chinese equivalent right?

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Yes, there are equivalents.   
fan edits = 粉丝(编/剪)辑 or 饭制(视频/剧)

Hope that helps :)

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edits编辑、剪辑.
fan edits饭制视频

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fan edits粉丝剪辑
edits,literally “编辑”,but we prefer to say “剪辑” in this situation.
The videos are usually called “饭制视频”或者“饭制剧”，

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粉丝编辑：粉丝可以去编辑或者剪辑，关于明星的文章，小视频等等。