  As a crucial step towards the completion of the fourth post-Newtonian (4PN)
gravitational-wave generation from compact binary systems, we obtain the
expressions of the so-called "canonical" multipole moments of the source in
terms of the "source" and "gauge" moments. The canonical moments describe the
propagation of gravitational waves outside the source's near zone, while the
source and gauge moments encode explicit information about the matter source.
Those two descriptions, in terms of two sets of canonical moments or in terms
of six sets of source and gauge moments, are isometric. We thus construct the
non-linear diffeomorphism between them up to the third post-Minkowskian order,
and we exhibit the concrete expression of the canonical mass-type quadrupole
moment at the 4PN order. This computation is one of the last missing pieces for
the determination of the gravitational-wave phasing of compact binary systems
at 4PN order.
