You may encounter this creature while you're grappling through the rooftops of Miagani Island. You need to catch up with it in the air and it's flying around buildings, making it hard to spot. It will occasionally shriek and will give you an approximate location and direction where it's heading. Just catch up with it and Batman will be able to take it down and get a blood sample.
Next, head to Chinatown to find Dr. Langstrom's lab. It's located inside an alley. Enter it and go down the elevator. Enter the lab,and inspect the body. Then access the computer nearby. You'll need to create an antidote for Langstrom's condition. You have to rotate the DNA sequence so that the DNA integrity will keep rising. An incorrect strain will decrease the integrity. Once the clean DNA sample has been synthesized, the section of this mission is complete.
Head to Bleake Island and keep your Detective Mode active as you scour the rooftops in search of the bat creature. You can also listen to the chatter about the sightings. Once you found it, chase after it and glide near enough to grab it and administer the drug. However, the dose doesn't seem to be enough so you have to wait a bit later until you can get new intel about this creature's whereabouts
Finally, the Man Bat will be flying around the Founders' Island skies. I happen to find him after the storyline with Harley Quinn. Like before, glide towards this creature to pummel him and administer a cure. Once the Man Bat is loaded, deliver it to GCPD to lock it up and complete the mission.