So if lye is what does the cleaning, why is it listed among like twenty ingredients, many of which are "cleaning agents" and "solvents," but it's function is listed as "pH adjuster?"

All Recipes "Decades ago, soaps were made with lye and vinegar, and they were too harsh for use on cast iron pans. They would indeed strip away oil and could remove seasoning. But today's soaps, especially ones that are made with eco-friendly solutions, are often too mild to remove seasoning."

Lodge "Fact: Soap isn't necessary, but you can use mild dish soap to clean cast iron. The seasoning on Lodge cast iron is fairly resilient and can withstand a little bit of soap, water, and a good scrub with a brush."

Wikipedia Experts advise against placing a seasoned pan in a conventional dishwasher.[17][18] While some food writers advise against all use of detergent for seasoned pans, tests by America's Test Kitchen found that small amounts of soap do not damage the seasoning.[19]

So in short: Yes there is lye in soap, no it isn't enough to affect cast iron, yes it's okay to use some soap to clean a cast iron pan.