The process of the writing of the verses of the Qur’an is clear, and it was done during the lifetime of the Prophet (P), and what happened after his demise was the compilation of the written texts.
A misunderstanding has arisen over the compilation of the Qur’an. Many westerners suppose that the Qur’an was written after the Prophet (P) by his companions like the Bible, which was written for the first time by Christ’s companions many years after God raised him to Himself. It means that the Bible was not recorded during the lifetime of Jesus. [1] Unlike the Bible, the Qur’anic verses were gradually revealed to the Prophet (P) in specific occasions during twenty three years, and there were some well-known companions who wrote the verses down exactly after their revelation. Since the Prophet (P) could not write and read, he ordered them to take dictation. Meanwhile, there were other well-known companions who memorized the very verses. [2] Memorization of the Qur’an is still prevalent among Muslims as a Prophetic tradition. Actually, they have confused the writing of the Qur’an with its collection. So what happened after the demise of the Prophet (P) was the unifying of these written codices of the Qur’an which were already recorded on the bone and skin of animals as well as on the hearts of many companions at the time of the Prophet (P). The words and verses of the Qur’an, that we have now, are the exact ones which were revealed to the Prophet (P); hence, neither omission, nor addition occurred.
[1] See: http://thebible.evangel.site/ & https://en.wikipedia.org/wiki/Bible#Old_Testament & https://www.thoughtco.com/when-was-the-bible-assembled-363293
[2] See: Ketab-o Al-Masahif, p 30; Tarikh Al-Qur’an, Zanjani Abu Abdullah, p 91; Manahil-o Al-Erfan, Zarqani, vol 1, p 246.