Place the grated papaya in a clean dish towel and twist the towel to squeeze out about half of the juice from the papaya, roughly ¼ cup of liquid. Discard the juice.
In a large saucepan, combine the papaya pulp, sugar, lemon zest, and lime juice. Bring the mixture to a simmer and cook, stirring, until it has a jellylike consistency, about 15 to 20 minutes.