 The arrow  formula_95  is the "inclusion of generators" set map from the set "GX" to the set "GFGX".The arrow  formula_96  is the set map from "GFGX" to "GX" which underlies the group homomorphism sending each generator of "FGX" to the element of "X" it corresponds to ("dropping parentheses").The composition of these maps is indeed the identity on "GX".