As the end step begins, Sygg's ability checks whether a player who is currently your opponent, or a player who was your opponent at the time he or she left the game, has lost 3 or more life over the course of the turn. If so, the ability will trigger. If not, it won't.

It doesn't matter how the opponent lost life or who caused it, as long as the total loss of life is 3 or more.

Sygg's ability checks only whether life was lost. It doesn't care whether life was also gained. For example, if an opponent lost 4 life and gained 6 life during the turn, that player will have a higher life total than he or she started the turn with -- but Sygg's ability will trigger anyway.

You don't specify an opponent when the ability triggers. If multiple opponents have lost 3 or more life over the course of the turn, the ability will trigger only once.

Sygg's ability will trigger even if Sygg wasn't on the battlefield at the time some or all of the life was lost.

Sygg's ability checks to see if it triggers at the end of every turn. It doesn't have to be your turn, and it doesn't have to be the turn of the opponent that lost life.