Electron-hadron colliders at the high-energy, high-intensity frontier would be the super-microscopes to resolve unambiguously parton dynamics using
deep inelastic scattering (DIS) in the TeV regime.
Adding a 60 GeV polarised electron beam onto the existing LHC and planned FCC-hh proton beams at CERN, a Large Hadron-electron Collider (LHeC) and a FCC-eh, with instantaneous luminosities of 10$^{34}$ cm$^{-1}$s$^{-1}$, could produce Higgs bosons abundantly and most importantly, also synchronously to any proton-proton data taking.
With an expected Standard Model Charged Current DIS cross section of around 200 (1000) fb at the LHeC (FCC-eh), the opportunities to study the dominant Higgs boson couplings with high precision are presented.
The compelling complementarity in measuring Higgs boson couplings at the LHeC (FCC-eh) and the respective hadronic HL-LHC (FCC-hh) colliders and between the potential $e^{+}e^{-}$ Higgs factories will be emphasised and the gain in the combined achieved accuracy highlighted.