You need to unlock this task on your own request, fortunately it is not a complicated process. At any moment of the game go to the Miagani Island and start pulling yourself up to the roofs of the nearby buildings by using the hook. After one of those actions Batman will have an unexpected encounter with a winged monster, which looks like a huge bat, and that will lead to adding a new mission.
Get a winged monster's blood sample
At the beginning of this task you need to locate the flying beast. In order to do that, you need to get to the center of the Miagani Island, bounce off the roof of one of the buildings and glide above the area. The loud squawk of the monster will help you locate it, as it will get louder when you get closer to it. What is more, use the detective mode as it will make the monster more visible to you.
After you've located the winged monster you have to get close enough to it so that Batman can catch it while still flying. In order to get closer to it, you need to use the hook to catapult Batman and to increase his flying speed. Glide towards the beast. After you've got close to the beast, the short video about catching a monster and getting its blood sample will turn on automatically (you don't need to interfere in any way).
Examine doctor Langstrom's laboratory
Listen to the conversation with Alfred, then go to doctor Langstrom's laboratory, which is located in Chinatown district on the Bleake Island. After getting there, get inside the building and use the elevator. You can watch the movie, which is displayed on the wall (this is optional). The most important thing is to examine the corpse of Francine Langstrom and to interact with a computer that belongs to a scientist, who turned into a huge bat.
There is a simple riddle to solve here. Depending on the needs you will have to either use one or two analog sticks (or suitable keys) to turn the DNA chains. You have to make sure that "clean", white chains are going through the rectangles that are on the picture above. After you've done it properly, the percentage of integrity will increase - it has to reach 100% for you to get the medicine.
Inject the medicine to the winged monster
The last part of that mission is very simple. Get out of the laboratory and start gliding above the central part of Bleake Island. Use the detective mode again in order to find the winged monster and get close to it while still flying. Batman will catch the creature and inject the medicine to it, but the final result won't be sufficient.
Next, get to the northern art of the Founders' Island and locate the winged monster for the last time. Use the gliding again in order to catch the monster and inject it with the medicine. All that is left for you to do is to take the monster to the GCPD Lockup and put it in the doctor Langstrom's (who, in meantime, will gain his human form back) solitary.