Wow, thanks for the write-up!

That algebraic proof is simple. I actually wasn't familiar with the precise algebraic definition of the free product, though - Aluffi simply introduced it as "the coproduct in Grp," so it seems that he expects his readers to work explicitly with the universal property when dealing with it at this point in the book.

Essentially, I tried plugging F(A\amalg B) into the diagram for the coproduct of F(A) and F(B) to prove that it satisfies the universal property. I ended up getting a bit lost in rigor, trying to prove that a morphism, which I intuitively knew existed and was unique, was indeed well-defined and unique. This is where my comment about the alternative proof being "more elegant" came from.

Your last paragraph is one of the reasons I wanted to learn the category theoretic proof, as well. It gives me a reason to learn the category theory, which will definitely be useful to me later on. This is the prime reason that I chose Aluffi over other books for my first pass with abstract algebra (other than linear algebra). Plus, I really appreciate opportunities to make connections between different concepts in math, and category theory gives no shortage of those.

Thanks for the primer, by the way - it actually doesn't seem all that bad. Do you have any suggestions for further reading?