Adjust A Bike Wheel’s Cup And Cone Rear Hub
Use The Left Side
An cup and cone hub has a lock-nut on the outside, then maybe a washer, then the cone which has two narrow flats. Cone spanners are thin spanners that fit onto the narrow flats. You need one that fits the narrow flats exactly this is likely to be somewhere between 13mm and 16mm.
Hold the axle still with an axle vice and turn the lock-nut anti-clockwise to unlock the bearing.
Adjust Rear Hub
Screw the cone in or out to tighten or loosen the bearing on the rear cup and cone hub. Make small adjustments and check the bearing between each.
Once the bearing on the rear hub is in the target zone of not-too-loose and not-too-tight hold the cone still and screw the lock-nut tight against it to lock the bearing.
Check the bearing on the rear hub and readjust if necessary. The act of locking may change the tightness of the bearing so if your first try at locking ends up too tight start again with the bearing very slightly looser.