  We present a comprehensive study of the light-cone generalized parton
distribution (GPD) and quasi-GPD of a flavor-neutral meson in the 't Hooft
model, {\it i.e.}, two-dimensional QCD (\QCDtw) in the $N_c\to\infty$ limit.
With the aid of the Hamiltonian approach, we construct the light-cone GPD in
terms of the meson's light-cone wave function in the framework of light-front
quantization, and express the quasi-GPD in terms of the meson's Bars-Green wave
functions and the chiral angle in the framework of equal-time quantization. We
show that, both analytically and numerically, the quasi-GPD does approach the
light-cone GPD when the meson is boosted to the infinite momentum frame, which
justifies the tenet underlying the large momentum effective theory for the
off-forward parton distribution. Upon taking the forward limit, the light-cone
and quasi-GPDs reduce to the light-cone and quasi-PDFs. As a bonus, we take
this chance to correct the incomplete expression of the quasi-PDFs in the 't
Hooft model reported in our preceding work [Y. Jia et al. Phys. Rev. D 98,
054011 (2018)].
