There are a few very misleading videos out there telling you that this is part of the main story and that you will just find the moon base on your map once completing it. Well this information is incorrect and i will show you exactly how to find the moon base and get the trophy. This will also lead you to the Lantern planets if you have been trying to find them also!
Yes this is after you complete the main story and you are in free play but what some videos creators are telling people in the comments is that they get the moon base on your map by completing the main story which is untrue. Like i say yes you complete the story but it will not just appear on your map once you do. Instead you need to follow my path on the video and you will find Daffy Duck dressed as the Green Lantern or it could be Duck Dodger if i remember my Daffy Duck cartoons as well as i do, either way once speaking to him for the first time you will be able build the gate just to the left of him, once that is done speak to him once more and then you are free to travel through the gateway and you will be on the moon base and then it will be on your map
Also as you see in the video, on the base are more gates, these are the gates to the Lantern planets.
I hope this clears things up for anyone who was also finding the same videos i had with incorrect information.
Happy gaming