I'm using a red LED grow light (36 watts, 660 nanometer wavelength) and the little plants are in a large white plastic container covered with a large clear plastic garbage bag to hold in the moisture. There is also a small open container of water inside next to the plants to help make sure the humidity levels within the enclosure do not go down. Occasionally I take the plants outside in the sun, but only when the temperature is above 70 °F, and for no more than 2½ hours at a time. (This is in the Pacific Northwest and right now the air temperatures are still pretty cool and fairly humid, there is no way I'd expose these young plants to any amount of full sun if they were in Southern California)
When I do take them outside, I always make sure to give them just a little bit of water before and right after to make sure the soil is moist. I do not plan on taking them outside later in the summer when the outside air starts becoming drier. 70 degrees is still fairly cool, but at 80 degrees I think there would be much more of a risk of drying out in the sun. Although it's true mangosteen likes higher temperatures, that is only if humidity levels are very high. I read the optimal minimum temperature for growing mangosteen starts at 77 °F. I think the small plants will grow faster in more light, but this is only assuming there is enough soil moisture and constant humidity to prevent them from drying out. Since these optimal conditions usually don't exist, I think that is the reason why small mangosteen plants usually cannot handle high levels of light.