Based on the…גמרא דף לט. “כל שבא (until the two dots) which is quite of the Mishna דף קמה עמ’ ב an item was cooked it may be reheated inside a Kli Rishon.
The Ran – no difference between solids or liquid ר”ן פרק ג. דף מג. “תנו רבנו מביא אדם קיתון של מים starting from the words “מיהו הני מילי בדבר שלא נתבשלה ” (towards the end of the paragraph)
Learn רשב”א דף לט. “חלח which says this Mishna is teaching us the rule אין בישול אחר בישול
Does this apply to liquid that was cooked and cooled down?
Review רש'”י דף לד. “גזירה שמא (three lines from bottom of the page) which states there is a violation of cooking by reheating item that was cooked and cooled down.
The רא”ש פרק ג ס’ י “ולמאי דפ’ דלא שייך בישול (last few lines) and ס’ יא gave a different explanation in the Gemara reason as to why one may not insulate an item on Shabbos as the Rabanan were concerned one might stoke the coals
Learn בית יוסף who discusses the machlokes between Rashi and the Ran whether reheating cooked liquids that have cooled down is a violation of cooking on Shabbos.
