Coadjutor engagement was an emergency step in the act of beaming. In taking this step, an engineer or transporter technician could be led to believe that the transporter was experiencing an event like a burnout. This step was preceded by the normal steps of locking onto and energizing a lifeform or object onto the transporter platform.
In 2266, after injuring himself on the surface of planet Alfa 177, geological technician Fisher was beamed aboard the USS Enterprise. During the beaming, the magnetic ore coating Fisher's jumpsuit interfered with the normal functioning of the transporter, necessitating Chief Engineer Montgomery Scott into taking this step. (TOS: "The Enemy Within")