Making vinegar is easy. At a simple chemical level, alcohol + bacteria from the acetobacter genus + oxygen + time -> acetic acid (vinegar).
There’s acetic acid bacteria floating around in the air, so red wine or cider left open over time will eventually turn to vinegar (the sugars in cider first fermenting to alcohol), but unpasteurized and unfiltered commercial vinegars may already contain “mother” (a significant amount of acetic acid bacteria + cellulose) that can be harvested to kick-start a new batch of vinegar (and ensure the acetic acid bacteria quickly becomes the dominant player and lowers the pH to a range where they are heavily favored).
In my case, I mixed a small bottle of organic unfiltered red wine vinegar and the remnants of a few bottles of left-open-too-long red wine, swirled/shaken together in a jar with cheesecloth over the top.
After two months of checking in, it finally tasted like a good red wine vinegar– so I tapped off two large bottles (one for now, one to age another year+ before using), and what’s left will be the starter for future bottles of unfinished red wine.