It sounds as if your insurance company is opening up an underinsured motorist property damage claim on your behalf because the at-fault party's liability property damage limits are not high enough to cover all of your damages.

Underinsured motorist property damage (UNDPD) is for when property damage is sustained by an insured and the negligent operator possesses insurance, but the limits of liability carried by the negligent driver are not sufficient to cover the damages.

If you don’t have Collision coverage, Underinsured Motorist Property Damage coverage pays up to a certain amount for repairs to the insured car (some states have limits at $3500, some are lower and some are higher). If you have Collision coverage, Underinsured Motorist Property Damage coverage only pays your Collision deductible (in some states).

This underinsured insurance comes into play once the at-fault party's limits have been reached, so their insurance is paying up to their limits and your underinsured motorist coverage takes over at that point.

You would have to speak to your insurance agent or claims adjuster to find out if your insurance carrier will later subrogate with the at-fault party to be reimbursed for these underinsured costs or not. Either way the at-fault party's insurance should be paying up to the person's limits so it is not only your insurance that is paying out on your total amount of damages.