This was during the lifetime of the prophet(saw) when every Hadith was directly written under his supervision.
the compilation started during the lifetime of the prophet (saw)
the compilation continued to the age of TABA-TABIEEN (next generation of companions) and in the subsequent ages.
The work never stopped and was a continuous process without any break in the chain process as orientalists wrongly state it.
The companions had 3 methods of learning and preservation of the hadith; memorization, writing, and practice.
The companions observed the actions of our Beloved Prophet (saw) and remembered it so applied it in their daily lives.
The companions wrote the Ahadith and memorized it.
Imam Abu Dawood narrated in his book that Hazrat Abdullah bin Umar" used to write whatever he heard from the prophet (saw).
however some companions forbade him to write as the prophet (saw) might say something out of anger or happiness which might not be part of the religion
The prophet (saw) said "By God who owns my life I say nothing but the truth; so you may right whatever you want to" .
All wrote Ahadiths:
Hazrat Abdullah bin Umar wrote more than 2000 Hadiths.
Hazrat Abdullah bin Amr
Hazrat Abu Hurairah
Hazrat Anas
Hazrat Ali
The prophet (saw) sent letters to emperors and kings therefore the companions took advantage and individually wrote and acted to what the Prophet (saw) said.