  In this article we investigate a monoid of smooth mappings on the space of
arrows of a Lie groupoid and its group of units. The group of units turns out
to be an infinite-dimensional Lie group which is regular in the sense of
Milnor. Furthermore, this group is closely connected to the group of bisections
of the Lie groupoid. Under suitable conditions, i.e. the source map of the Lie
groupoid is proper, one also obtains a differentiable structure on the monoid
and can identify the bisection group as a Lie subgroup of its group of units.
Finally, relations between groupoids associated to the underlying Lie groupoid
and subgroups of the monoid are obtained. The key tool driving the
investigation is a generalisation of a result by A. Stacey which we establish
in the present article. This result, called the Stacey-Roberts Lemma, asserts
that pushforwards of submersions yield submersions between the
infinite-dimensional manifolds of mappings.
