When the Messenger of Allah ﷺ had prayed the two units of the dawn prayer, he would lie down on his right side.
كان رسول الله ﷺ إذا صلى ركعتي الفجر، اضطجع على شقه الأيمن.
Bukhari narrated it in the chapter on tahajjud (1160) from Abdullah bin Yazid, who said, Saeed bin Abu Ayub told us, who said, Abu Aswad told me, from Urwa bin Zubayr, from Ayesha, and he mentioned something similar.Muslim narrated it (736) with another chain, from Amr bin Harith, from Ibn Shihab, from Urwa bin Zubayr, from Ayesha, the wife of the Prophet ﷺ, who said, "The Messenger of Allah ﷺ used to pray eleven rakats between the time he finished the Isha prayer, which people call Atamah, and Fajr, giving the greeting of peace after every two rakats, and finishing with a single rakat as witr. When the muezzin fell silent after the call for the Fajr prayer, and dawn had become clear to him, and the muezzin came to him, he would get up and pray two light rakats, then he would lie down on his right side until the muezzin came to him for the call to begin the prayer."