Yes when you go in for surgery and are put under for the surgery the anesthesiologist will remain in the surgery room at your side for the surgery to help ensure you're breathing properly and staying under during the surgery.
The anesthesiologist will be there in surgery in case you start to wake up as well.
During surgery there will be 2 to 4 surgery techs in the surgical room to ensure the surgery goes smoothly so it's not just one or two surgeons in the operating room.
The anesthesiologist will be there the whole time during surgery to help the surgeons out to ensure that you remain asleep and to make sure you're breathing properly as well.
They want to make sure everything goes properly and that you come out of the surgery alive.
Some people do die during or after surgery but most surgeries go very smoothly with the person recovering fully.