  We propose a notion of a proper Ehresmann semigroup based on a
three-coordinate description of its generating elements governed by certain
labelled directed graphs with additional structure. The generating elements are
determined by their domain projection, range projection and $\sigma$-class,
where $\sigma$ denotes the minimum congruence that identifies all projections.
We prove a structure result on proper Ehresmann semigroups and show that every
Ehresmann semigroup has a proper cover. Our covering monoid turns out to be
isomorphic to that from the work by Branco, Gomes and Gould and provides a new
view of the latter. Proper Ehresmann semigroups all of whose elements admit a
three-coordinate description are characterized in terms of partial multiactions
of monoids on semilattices. As a consequence we recover the two-coordinate
structure result on proper restriction semigroups.
