I've started to read the book Trees by Jean Pierre-Serre, and in Its first section he defines the direct limit of groups. I'm pretty used to the canon definition of direct limits which requires a directed set of indexes $I$ and a set of morphisms satisfying certain conditions (see for example this). The Serre's definition (see below) does not need a directed (nor even an ordered) set of indexes, and it usually appears as a consequence of the definition I know (at least in the basics of group theory, because I know that that actually comes from a more general notion in category thoery). What I really can't handle is his construction which is based on generators and relations.
I would really appreciate any ideas that could help me to understand that construction and why (if) it is equivalent to the usual one.
Thanks in advance.