Sleeving is a way to extend the life of otherwise valuable double guns with worn or damaged barrels.
To sleeve a double gun, the old barrels are cut off just ahead of the chambers, and the chambers are reamed out. The original parts that attached to the action - hook, barrel lumps, extractors, etc. - are preserved. New barrels, with the outside of the new chambers turned down to fit, are fitted into the original reamed out old chambers. Done well, the seam between old and new sections is almost invisible.
The result is something like the monoblock construction favored by some modern gunmakers - brand new, strong barrels inserted into the old gun.