We show that the universal X-generated F-inverse monoid F(G), where G is an X-generated group, introduced by Auinger, Szendrei and the first-named author, arises as a quotient inverse monoid of the Margolis-Meakin expansion \(M(G, X\cup {\overline{G}})\) of G, with respect to the extended generating set \(X\cup {\overline{G}}\) , where \({\overline{G}}\) is a bijective copy of G which encodes the m-operation in F(G). The construction relies on a certain dual-closure operator on the semilattice of all finite and connected subgraphs containing the origin of the Cayley graph \({\text {Cay}}(G, X\cup {{\overline{G}}})\) and leads to a new and simpler proof of the universal property of F(G).
