pictures of 2 lychee trees, south of LA, a Mauritius and the little one is a Sweetheart
They don't seem to mind full sun, and only took a lot of water the first year when their root systems were becoming established. I'd say they only require a medium level of water now, and can go for a week without being watered (maybe not in the summer though).
Little lychee seedlings have much more demanding needs though, soil needs to be kept constantly moist, and too much sun will dry them out, while they will not grow with too little sun, very hard to grow a lychee from seed into a tree in this climate (though the seeds do sprout easily).
The lychee trees appear to be very slow growing, so patience will be needed. There are already some tiny little bloom pannicles on the bigger tree.