How do I install the bottom tee?
Your first paragraph ...Tee Instructions: You first install the “Tee” to the liner using the band that is attached. When installing the liner from the top down, have another person on the bottom waiting for the liner to arrive. When the liner is visible, the bottom person will insert the “snout” of the tee into the thimble. This will require good communication. The 2 people will now work together lining up the hole on the tee with the snout. The most important part of the installation is that the lips of the tee are inside the snout. If your chimney has a bend in it or the liner is having trouble going down the chimney, one option is installing the tee on the bottom in the chimney. This will require more work and skill but can be done. Sometimes this is not an option, every situation is different. You will need to open up a hole large enough in the chimney near the thimble to install the tee. Many times the extra work of breaking a hole and patching it is less work than fighting with the liner to get it down. But it is up to you.
If you go with this option, make sure the end of liner is in good condition, meaning it is nice and round and makes a snug fit into the tee. There should be no gaps between the liner and the tee.
If you have a two appliances you want to connect to one chimney liner, you would need to connect the two appliances together before entering the liner. This means using a stove pipe "Tee". They are available and any plumbing store. They will have 3 outlets, most commonly a 6x6x63. This will allow a 6" furnace to connect to the snout on the liner and have a 3" water heater venting into the same chimney liner.
After everything is connected and secure you can fill the hole around the snout with brick and mortar. Complete your connection to the heating appliance following applicable codes.