  We consider the crossed product $G_{ga}$ by $\R_+^*$ of the adiabatic
groupoid associated with any Lie groupoid $G$. We construct an explicit Morita
equivalence between the exact sequence of order 0 pseudodifferential operators
on $G$ and (a restriction of) the natural exact sequence associated with
$G_{ga}$. As an important intermediate step, we express a pseudodifferential
operator on $G$ as an integral associated to a smoothing operator on the
adiabatic groupoid $G_{ad}$ of $G$.
