  Recently, a 80 K superconductor was discovered in La$_3$Ni$_2$O$_7$ under
high pressure. Density function theory (DFT) calculations identify
$d_{x^2-y^2}$, $d_{z^2}$ as the active orbitals on the bilayer square lattice
with a $d^{8-x}$ configuration of of Ni per site. One naive expectation is to
describe this system in terms of a two-orbital t-J model. However, we emphasize
the importance of Hund's coupling $J_H$ and the $x=0$ limit should be viewed as
a spin-one Mott insulator. Especially, the significant Hund's coupling shares
the inter-layer super-exchange $J_\perp$ of the $d_{z^2}$ orbital to the
$d_{x^2-y^2}$ orbital, an effect that cannot be captured by conventional
perturbation or mean-field approaches. In this study, we first explore the
limit where the $d_{z^2}$ orbital is Mott localized, dealing with a one-orbital
bilayer t-J model focused on the $d_{x^2-y^2}$ orbital. Notably, we find that
strong inter-layer pairing survives up to $x=0.5$ hole doping driven by the
transmitted $J_\perp$, which explains the existence of a high Tc superconductor
in the experiment at this doping level. Next, we uncover the more realistic
situation where the $d_{z^2}$ orbital is slightly hole-doped and cannot be
simply integrated out. We take the $J_H\rightarrow +\infty$ limit and propose a
type II t-J model with four \textit{spin-half} singlon ($d^7$) states and three
\textit{spin-one} doublon ($d^8$) states. Employing a parton mean-field
approach, we recover similar results as in the one-orbital t-J model, but now
with the effect of the $J_\perp$ automatically generated. We propose future
experiments to electron dope the system to further enhance $T_c$.
