The best way to tell if the chimney is the right type is to look for a label. Usually the installer will buy a package with the stove and matching chimney. The stove label may state which chimney is recommend but not always. The chimney for a wood stove must be rated to 650 degrees C or 2100 degrees F. The label on the chimney should say it is certified and complies with the S629 standard within the B365 of the building code. More often than not the label is missing or not open where you can see it. Another approach is to try to see how thick the insulation is between the two layers of the chimney. Most wood stove chimneys will have 2” of insulation. There are a couple exceptions but if the chimney only has 1” of insulation it is most likely only rated to 1700 degrees and only acceptable for use on a wood fireplace. As an installer I generally can tell by looking at the chimney if it is safe for use on a wood stove or not. I recommend having the entire setup inspected for your safety.