How to Make Up Fajr Prayer and Its Intention
For Muslims who miss the Fajr prayer because they overslept or forgot, Islamic law makes it easy by allowing them to make up the prayer as soon as they wake up or remember. The intention for making up Fajr: أُصَلِّي فَرْضَ الصُّبْحِ رَكْعَتَيْنِ مُسْتَقْبِلَ الْقِبْلَةِ قَضَاءً للهِ تَعَالَى (Usholli fardhos subhi rok'ataini mustaqbilal qiblati qodho'an lillahi ta'ala), which means 'I intend to make up the Fajr prayer, two rak'ahs, facing the qibla, for the sake of Allah Almighty.'
The movements and pillars of the makeup prayer are the same as a regular Fajr prayer, which is two rak'ahs. The only difference is in the intention. Regarding how loud the recitation should be, some scholars say it follows the time you're making it up (quietly if it's daytime, loudly if it's nighttime), while another opinion says you should stick to the original nature of Fajr prayer, which is jahar (recitation said out loud).
The legal basis for making up prayers is found in a hadith: 'Whoever misses a prayer because of sleep or forgetfulness, then let him perform it when he remembers. There is no penalty for it except that.' (Narrated by Bukhari). In the Quran, Surah Al-Isra' verse 78 also stresses the command to guard the Fajr prayer. The Prophet Muhammad (peace be upon him) and his companions even experienced this once when returning from the Battle of Khaibar, and then he made up Fajr in congregation after they woke up.
https://mozaik.inilah.com/ibad