Personally instead of using soap as part of the seasoning process, I pour a bunch of Morton sea salt into the pan and a little bit of water, and abrade it that way with a sponge. It also has the benefit of creating a lovely matte finish at the end of the seasoning process. I've also found the seasoning process works better at each iteration this way.
I clean the pan about 80% of the time with just salt and a paper towel. Pour salt in, rub it around, dump salt in the trash, marvel at how nice the pan looks (seriously, I have to admire it every single time).
The other 20% I need water because something stuck bad, and usually I'll just use my soapy sponge. But if the pan is real hot still, I'll just put water in it and let it deglaze the stuck stuff.
It's also how I clean the pans if I have to.