The Prophet (ﷺ) used to lie down on his right side, after offering two rak`at (Sunna) of the Fajr prayer.
Sahih al-Bukhari Book 19, Hadith 39
← Previous Hadith Allah's Messenger (ﷺ) offered the `Isha' prayer (and then got up at the Tahajjud time) and offered eight rak`at and then offered two rak`at while sitting. He then offered two rak`at in between the Adhan and Iqama (of the Fajr prayer) and he never missed them.

Sahih al-Bukhari Book 19, Hadith 38 Next Hadith → After offering the Sunna of the Fajr prayer, the Prophet (ﷺ) used to talk to me, if I happen to be awake; otherwise he would lie down till the Iqama call was proclaimed (for the Fajr prayer).

Sahih al-Bukhari Book 19, Hadith 40