To see how this relates to the constructive definition, think of the mapping from "S" to "FS" as sending each symbol to a word consisting of that symbol. To construct "φ" for the given , first note that "φ" sends the empty word to the identity of "G" and it has to agree with on the elements of "S". For the remaining words (consisting of more than one symbol), "φ" can be uniquely extended, since it is a homomorphism, i.e., "φ"("ab") = "φ"("a") "φ"("b").