Hot water on shabbos

Question

If hot water, say 150 degrees, is taken from a hot water urn on shabbos, then quickly put into a second cup (kli sheni), it is permissible to use the water for almost anything (i believe even cooking, e.g a tea bag). however, in the kli sheni the water is still above 125 degrees, so this would really be cooking which is assur on shabbos. So why is it good enough to transfer the water to a kli sheni?? Please list sources for this if there are any Thanks SM