"al umm al Shafi'i" is a book attributed to imam al Shafi'i.
But everything in the book is "al shafi'i said: ...." so clearly he himself did not write the book. Some articles say it was dictated to his student. But that can't be true either because in many parts of the book al shafi'i is the second or third person in a chain. for example:
أَخْبَرَنَا الرَّبِيعُ قَالَ أَخْبَرَنَا الشَّافِعِيُّ
So who really wrote this book? thanks
Imam a-Shafi'i wrote the book, but it was not copied from him, rather it was transmitted (orally) to us via his students one of these students was a-Rabi' ibn Sulayman al-Muraadi الربيع بن سليمان المرادي, who was among the last who narrated from a-Shafi'i and for this reason is the main source for a-Shafi'i's work on his new-Madhhab (his madhhab when a-Shafi'i moved to Egypt and settled there).
Some people attributed this book to a-Rabi' because he divided his teachers work into chapters and arranged it. However actually it was another student Yusuf ibn Yahya al-Buwayti يوسف بن يحيى البويطي who collected the book which is a collection of all a-Shafi'is works on Fiqh. The book was compiled in Egypt where a-Shafi'i setlled 199 a.H. and died 204 a.H.
So basically the book was originally written by imam a-Shafi'i and its content was re-collected by al-Buwayti whom left it to ar-Rabi' -as a narrator- to re-arrange it. In other words The book we have by now is the narration of a-Rabi' (who should know the content from a-Shafi'i as he was his student) from -the manuscript of- al-Buwayti from -the lessons, papers, manuscripts of- a-Shafi'i. As stated in the linked source which also explains why al-Buwayti left it to a-Rabi' to narrate this book:
وقد قام البويطيّ تلميذُ الإمام الشافعي بجمعه دون أن يذكر اسمه فيه، ولا نسبه إلى نفسه, بل نُسِب إلى راويه الربيع بن سليمان المرادي (174-270ﻫ) تلميذ الإمام الشافعي أيضاً، والذي بوبه ورتبه وهذبه.
Al-Buwayti, a student of Imam al-Shafi’i, compiled it without mentioning his name in it, nor attributing it to himself. Rather, it was attributed to its narrator, al-Rabi’ ibn Sulayman al-Muradi (174-270 AH), a student of Imam a-Shafi’i as well, who wrote, arranged and refined it.
قال الحافظ أبو الحسين محمد بن عبد الله بن جعفر الرازي المتوفى سنة 347ﻫ، وهو والد الحافظ تمام الرازي: كان البويطي يقول: الربيع أثبت في الشافعي مني. التهذيب للحافظ ابن حجر (3: 246).
Al-Hafiz abu al-Hussain Muhammad ibn 'Abdullah bin Ja'afar ar-Razi, who died in the year 347 A.H., and he is the father of al-Hafiz Tammam ar-Razi, said: Al-Buwayti used to say: a-Rabi' is more correct in his narration from a-Shafi’i than me. A-Tahdheeb by Al-Hafiz ibn Hajar (3: 246).
So one could say a-Rabi' did a proof reading of the book therefore it can be attributed to him as the last narrator or we could say it is a-Shafi's al-Umm on the narration of a-Rabi'.
Here you may find an introduction on al-Umm -in Arabic-
