This is a solution manual for ALGEBRA, Chapter 0, an algebra textbook written by Paolo Aluffi. The way of introducing abstract algebra in a language of categories appeals to me very much. So I'm perfectly willing to share my solutions to the exercises in this book.
As a beginner, I'm trying to make my proof explict and detailed, but unfortunately it seems verbose and tedious sometimes. Anyway, I would appreciate it if you can point out my mistakes, big or small.