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

$E$-unitary and $F$-inverse monoids, and closure operators on group Cayley graphs