As the title suggests, I'm confused with the definition of a free object, and would be thankful if someone could present some illustrative examples and motivation for understanding the definition. Let F be an object in a concrete category C, X a nonempty set, and i : X --> F a map (of sets). F is free on the set X provided that for any object A of C and map (of sets) f : X --> A, there exists a unique morphism of C, f* : F --> A, such that f*i = f. Ok, first of all, since concrete categories are mentioned, one is dealing with mappings between underlying sets. So, what does the deifnition actually tell? Is the emphasis on the fact that for every mapping f, we can "simulate" this very mapping with f*i? I believe it is a bit hard to understand such definitions without examples (the *one* in the book didn't really help me) and greater experience in mathematical abstraction.