Let be a connected Lie group and a generating Lie semigroup. An important fact is that generating Lie semigroups admit simply connected covering semigroups. Denote by the simply connected universal covering semigroup of . In connection with the problem of identifying the semigroup of monotonic homotopy with a certain subsemigroup of the simply connected covering semigroup we consider in this paper the following subsemigroup where is the lifting to of the exponential mapping . We prove that is also simply connected under the assumption that the Lie semigroup is right reversible. We further comment how this result should be related to the identification problem mentioned above.