The LHeC is a new $ep$ collider of unprecedent kinematic range,
luminosity and precision in deep inelastic scattering.
This leads to the first ever complete measurement
of PDFs, including, for example, the strange density.
It furthermore extends to very high Bjorken $x$ in $ep$ and
to so large $Q^2$ that no nuclear or higher twist
corrections affect the high $x$ PDF determinations.
A much deeper understanding of quark-gluon dynamics
is needed and  in sight, exemplified by the potential of the
LHeC to measure $\alpha_s(M_Z^2)$ to per mille accuracy.

The LHeC is designed to operate synchronously with the HL-LHC.
Preparations have begun, following the detailed 
design concept report and a corresponding mandate
from CERN, supported by ECFA and NuPECC, to prototype
critical components and to carry out more detailed studies,
as of the interaction region and high luminosity optics,
in major international collaborations with CERN. As
has been indicated here, there is a potential
for the LHeC to achieve luminosities in excess
of $10^{33}$\,cm$^{-2}$s$^{-1}$. The formation of a
detector collaboration has indeed begun.

The present paper discussed the relation of the LHeC
to the LHC, in its HL phase. As indicated by the 
ATLAS contribution to the ESPP, there are two major
questions for the HL-LHC to investigate, the properties
of the 126\,GeV boson, which likely is the Higgs particle,
and the search for maximum mass particles, of several
TeV in direct production mode. It has been argued here
that the LHeC can assist the transformation
of the  LHC into a precision  Higgs facility,
 with its own coupling
and CP measurements based on clean $WW$ fusion in $ep$,
and with its ultra-precise PDF, heavy quark and
$\alpha_s$ determinations, which will reduce
the theoretical uncertainties of Higgs measurements
in $pp$ to a negligible level. It is similarly 
apparent that the discovery potential of new
particles at high masses at the HL-LHC is severely
limited by the deviations and uncertainties of
the predictions based on currently available
PDF sets and as well the uncertainties of
input parameters such as the heavy quark masses 
and $\alpha_s$.
For the Higgs the prominent decay
$H \rightarrow b \overline{b}$ and for SUSY gluino
pair production have been used as first representative
examples. 

It has been also discussed which role
DIS and Drell-Yan scattering can play in the 
determination of PDFs, which may be illustrated
by comparing the HERA and Tevatron PDF related results.
Obviously $ep$ provides precision information
on the quark structure and quark-gluon dynamics
while the prime task of $pp$ is the extension
of the energy frontier. With the Higgs measurements and
BSM searches at the LHC, these aspects appear intimately 
connected, stronger than before, because both
$pp$ and $ep$, owing to higher energies and
huge luminosities, can explore the high mass
limits of phase space.

The paper has also summarised the importance 
of electron-ion scattering for the LHC heavy ion
physics programme, with the determination of nuclear PDFs
in a phase space extended by $4$ orders of magnitude
and diversified with heavy quark nPDFs to be measured
for the first time. The LHeC determines the
initial state of the QGM and leads to a quantitative
understanding of the hadronisation in media, controlled
by the electron DIS kinematics.

The LHeC has its own, fundamental physics programme,
as with saturation and diffractive physics, with the
resolution of deuteron, neutron and photon structure,
for example. Its relation to the LHC extends beyond
the nowadays most prominent examples, Higgs measurements
and SUSY searches, as with top, leptoquark, excited
lepton or electroweak physics. The CDR has demonstrated
that the LHeC may be realised without any major
%interrupt of the GPD programme or 
extra machine delays.
Pairing the unique hadron beams of the LHC with
a new lepton beam would therefore substantially
enrich the physics potential of the LHC facility and make
optimum use of the major investment already made
and envisaged for the LHC.

