Why is there still a drain tube in my cat's abscess?

I recently to my cat to the vet after I noticed a nasty abscess under his chin. The vet gave him the normal check up and gave him his shots. He said that he would need to keep my cat overnight to sedate him to drain the abscess. I signed the permission form and wished for my baby to get well. The next day, I called to see if he was finished and they told me he was ready to be picked up. I picked him up and put him in his own room when I got home. The vet also gave me an amoxcillin treatment to give my cat by mouth twice a day. He's been doing fine since the surgery, but today I noticed his wound and I saw the stitches and the drain tube was still there. Is this tube supposed to stay in after surgery or were they supposed to remove it? If it's supposed to stay in, does it fall out later on it's own?