  We study the capabilities of a muon collider, at 3 and 10 TeV center-of-mass
energy, of probing the interactions of the Higgs boson with the muon. We
consider all the possible processes involving the direct production of EW
bosons ($W,Z$ and $H$) with up to five particles in the final state. We study
these processes both in the HEFT and SMEFT frameworks, assuming that the
dominant BSM effects originate from the muon Yukawa sector. Our study shows
that a Muon Collider has sensitivity beyond the LHC, as it not only relies on
the Higgs-decay branching fraction to muons. A 10 TeV muon collider provides a
unique sensitivity on muon and (multi-) Higgs interactions, significantly
better than the 3 TeV option. We find searches based purely on multi-Higgs
production to be particularly effective in probing these couplings.
