  We show that the category of $X$-generated $E$-unitary inverse monoids with
greatest group image $G$ is equivalent to the category of $G$-invariant,
finitary closure operators on the set of connected subgraphs of the Cayley
graph of $G$. Analogously, we study $F$-inverse monoids in the extended
signature $(\cdot, 1, ^{-1}, ^\mathfrak{m})$, and show that the category of
$X$-generated $F$-inverse monoids with greatest group image $G$ is equivalent
to the category of $G$-invariant, finitary closure operators on the set of all
subgraphs of the Cayley graph of $G$. As an application, we show that
presentations of $F$-inverse monoids in the extended signature can be studied
by tools analogous to Stephen's procedure in inverse monoids, in particular, we
introduce the notions of $F$-Sch\"utzenberger graphs and $P$-expansions.
