The question arises whether it is permissible to warm up cold water in a cup by adding hot water from an urn on Shabbat.
The Shulhan Aruch (siman 318:11) discusses a case of "Ambati"-a bathtub filled with hot water. He rules that one may not pour cold water into it, since it will be cooked by the hot water. However, one may pour hot water from the bath into cold water. The question is what the status of this "Ambati" is. The Rema, in parentheses, adds that the Ambati is a "Keli Sheni"-not the original vessel in which the water was heated. According to this understanding, Maran is teaching that although the Ambati is a Keli Sheni, nevertheless, it is hot enough to cook cold water poured into it, but does not cook when poured into cold water.
However, the Rambam, who is the source for Maran, uses the same language as Maran, and the Kesef Mishneh (22:5-6) understands him to mean that the Ambati is a Keli Rishon. Accordingly, the Hidush is that one may pour hot water from a Keli Rishon into cold water. This is based on the opinion of Bet Hillel in Masechet Shabbat that "Tatah Gavar"-whatever is on the bottom overpowers, which in this case is the cold water. That is, the hot water does not cook the cold water, but the cold water chills the hot water.
Hacham Ovadia, in Yabia Omer, writes that, in principle, Sepharadim would follow this lenient understanding. However, Tosafot add a caveat that the cold water only overpowers the hot water poured into it, when there is a lot of cold water. Even though the Rambam does not distinguish, Hacham Ovadia and Hacham David in Halacha Berura conclude that one should not be lenient unless there is a significant amount of cold water, and one is just adding a small amount of hot water.
The Hafetz Haim, in his Be’ur Halacha (318:11-12), discusses this in the context of preparing tea "Sense"- concentrated liquid essence of tea leaves. This Halacha is more lenient in this case and permits pouring hot water from the urn onto the cold tea essence, since there are three lenient factors. First, the tea essence was already cooked, and there is an opinion that "En Bishul Ahar Bishul B’Lach"-there is no problem of cooking even a cold liquid that was already cooked. Second, there is also the leniency of the Kesef Mishna’s understanding of the Rambam that "Tatah Gavar" even when pouring hot water from a Keli Rishon. Third, there is also an opinion that "Irui"-pouring hot water-never constitutes cooking on Shabbat. With these three factors, it is permitted for Sepharadim to pour hot water on tea essence, even if the cold tea is a minute quantity. One need not take the opinion of Tosafot into account.
One may pour hot water from an urn into cold water, only if there is a significant amount of cold water. However, it is permitted to pour even a large quantity of hot water into a small amount of tea essence.