  Large Hadron Collider (LHC) has provided direct evidence of Yukawa couplings
between the third generation charged fermions and the 125 GeV Higgs boson.
Whether the first two generation charged fermions arise from exactly the same
mechanism becomes the next interesting question. Therefore, direct measurements
of charm or muon Yukawa couplings will be crucial to answering this puzzle. The
charm Yukawa measurement at the LHC suffers from severe QCD background and it
is extremely difficult to reach the sensitivity. In this paper, we compare the
potential of probing charm Yukawa coupling at the two proposed future "Higgs
Factory" experiments, the Large Hadron electron Collider (LHeC) and Circular
electron positron collider (CEPC). At the LHeC, Higgs bosons will be produced
via weak boson fusion and the energetic forward jet may suppress the background
significantly. However, due to huge $\gamma-g$ scattering background, the
potential of LHeC search is still limited. With $-80\%$ polarized electron beam
of 60 GeV, the signal significance can only reach $2\sigma$ for $
\kappa_{c}\simeq 1$ with a 3 ab$^{-1}$ integrated luminosity. In comparison,
measurement at the CEPC can then reach $8.0\sigma$ for $\kappa_{c}\simeq 1$
with a 2 ab$^{-1}$ of data.
