Yes.  Each coffin or casket is identified on arrival at the crematorium and the identification label or nameplate is placed on the outside of the cremator as soon as the coffin is placed inside.  The label or nameplate stays there until the remains are removed and it is then transferred with the remains to the cooling tray, which then proceeds to the preparation room.  The label or nameplate stays with the remains until they are placed in a container, which is also clearly identified.  In addition, each cremation chamber will only accept one coffin or casket.  This ensures all cremated remains are kept separate throughout the entire cremation process.  The containers are then securely stored awaiting disposition instructions.  Contact your local crematorium for further information as to their operations.