Tumaiyowit and Mukat are the two quarreling gods who ruled the island at the beginning of the world. They're a part of the island's mythology. According to Karana, "Tumaiyowit wished people to die. Mukat did not" (12.8). Tumaiyowit went down to the underworld, and hence that's where people go when they die.
The gods represent two different ways of seeing life on the island and different ways of thinking about life. Which god is Karana more like?