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.