When Jay Sweetwatter dies, another River Rat assumes this role, and takes on the name. Their old self is given a funeral and is assumed dead (cause of death is murdered by Sweetwater Jay of course). From that point on, all River Rats will claim that this incarnation of Jay is the true one, and furthermore the only one that has ever existed. Anyone claiming otherwise is either flogged or dragged under the keel.
Each incarnation of Jay traditionally tries to imitate their predecessor's mannerisms to the best of their ability, but embellishes them with their own flourish. Current version of Jay is a lout and feisty human red-head who drinks hard and fights with a rapier. She is fairly young as far as Jay's incarnations go, but sometimes the fearless leader must be replaced on short notice, and best qualified member of the gang present at the moment must take on the mantle.