My cat is gagging, throwing up, and coughing. Should I see a vet?

Hello and thank you for reaching out to Petco Pet Education Center, formerly Petcoach. I'm sorry to hear that Mittens has been vomiting and is now coughing. Cats can vomit for a number of reasons, including trying to vomit a hairball, eating something they shouldn't have, gastrointestinal obstruction, infection, inflammation, or even cancer. From the information you provided, my concern is that Mittens may have aspirated (taken fluid into the lungs) during one of these vomiting episodes, and is now coughing as a result. Aspiration pneumonia can be a scary thing in cats, and especially if Mittens is not acting himself or wanting to eat, I would recommend bringing him to your veterinarian to be examined. If your veterinarian determines that he has abnormal lung sounds, he/she may recommend blood work and chest x-rays to look for evidence of pneumonia. If present, they will likely start Mittens on some antibiotics. There are other possibilities for Mitten's cough including bronchitis or an upper airway infection, which should also be managed by a veterinarian. I hope this is helpful!