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1. The Mitzvah to eat hot food on Shabbos:
It’s a Mitzvah to insulate food for Shabbos in order so one can eat hot food on Shabbos, as [eating hot food] is [part of the mitzvah] of honoring and having enjoyment on Shabbos. [See Q&A 1]
The law by one who hot food is unhealthy for him: However, for one whom eating hot food causes him damage, it is permitted for him to eat [only] cold foods. However, one who does not believe in the words of the Sages and prohibits eating hot food on Shabbos, we suspect that perhaps he is a heretic.
It is a Mitzvah to insulate food on Erev Shabbos in the permitted ways to be explained so one can eat hot food on Shabbos.
Must one specifically insulate his food, or is may he leave it on the fire from Erev Shabbos?
The main point is that the food stays hot for Shabbos. Thus, it makes no difference if one achieves this through insulation or through leaving it over a flame.
Insulating food on Erev Yom Kippur so it will be ready to eat after the leave of Yom Kippur:
First Opinion: There are those that say that insulating on Erev Yom Kippur for after Yom Kippur is forbidden being that it’s considered that Yom Kippur is preparing for a weekday, which is forbidden to be done, and insulation was only allowed to be done in honor of Shabbos.
Other Opinions: However, others hold that only if one does an action on Yom Kippur to cause it to prepare for a weekday is it forbidden, however if it occurs on its own then there is no prohibition.
Final Ruling: Although we rule like this latter opinion, nevertheless the custom is to be stringent like the former opinion.
May one leave food cooking over a fire on Erev Yom Kippur for the sake of eating on Motzei Yom Kippur?
Based on the above custom one may not do so.
May one leave food cooking over a fire on Erev Yom Kippur for the sake of children or an ill person?
Yes. In such a case it is permitted to leave food over the fire, or insulate food before Yom Kippur, whether they plan to eat the food on Yom Kippur or desire to have the food ready to be eaten immediately after Yom Kippur.
May one insulate foods with material that does not add heat for the need of Motzei Shabbos?
May one insulate foods with material that adds heat for the need of Motzei Shabbos?
 Admur 257:13
 As the apostate Tzedoki clan which did not believe in the Oral Torah held that one may never eat hot food on Shabbos. Thus, we suspect that one who follows this is part of this clan of heresy.
 Shabbos Kehalacha Vol. 2 page 1
 Admur 609:1
 What honor of Shabbos is there to allow insulating food on Erev Shabbos for after Shabbos? Perhaps it is referring to insulating food on Erev Shabbos for Shabbos, and since the sages allowed one to insulate for Shabbos they did not want to differentiate and thus also allowed insulating for after Shabbos.
 Piskeiy Teshuvos 609:1; See Shabbos Kehalacha 2 11:27 [p. 48-51]; However see M”A 257:2 that “One should not add to the Issur of insulating from before Yom Kippur”
 Alef Hamagen 609:1; Piskeiy Teshuvos 609:1
 See Shabbos Kehalacha 2 11:27 [p. 48-51]
 M”A 257:2 based on Rama 609:1
 Elya Raba; Levush; M”B 257:4; Shaareiy Tziyon 609:1; Piskeiy Teshuvos 257:5; See Shabbos Kehalacha 2 11:27 [p. 48-51]
Ruling of Admur: Admur omits the entire discussion of the M”A ibid and does not relate either way to the law of Motzei Shabbos. This implies that he does not hold of any prohibition involved. [Shabbos Kehalacha ibid p. 51]
 Piskeiy Teshuvos 253:5; See Admur 254:11; M”A 254:20; Machatzis Hashekel ibid; Shabbos Kehalacha 2 11:27 [p. 48-51