  We prove that four different notions of Morita equivalence for inverse
semigroups motivated by, respectively, $C^{\ast}$-algebra theory, topos theory,
semigroup theory and the theory of ordered groupoids are equivalent. We also
show that the category of unitary actions of an inverse semigroup is monadic
over the category of \'etale actions. Consequently, the category of unitary
actions of an inverse semigroup is equivalent to the category of presheaves on
its Cauchy completion. More generally, we prove that the same is true for the
category of closed actions, which is used to define the Morita theory in
semigroup theory, of any semigroup with right local units.
