  To a Lie groupoid over a compact base, the associated group of bisection is
an (infinite-dimensional) Lie group. Moreover, under certain circumstances one
can reconstruct the Lie groupoid from its Lie group of bisections. In the
present article we consider functorial aspects of these construction
principles. The first observation is that this procedure is functorial (for
morphisms fixing the base). Moreover, it gives rise to an adjunction between
the category of Lie groupoids over a fixed base and the category of Lie groups
acting on the base. In the last section we then show how to promote this
adjunction to almost an equivalence of categories.
