Interpretation of Coulomb operator in Hartree-Fock equation

I have read in a textbook (Modern Quantum Chemistry Szabo and Ostlund) that the Coulomb operator of the form
begin{equation}
mathcal{J}_{j}left(mathbf{x}_{1}right)=int d mathbf{x}_{2}left|chi_{j}left(mathbf{x}_{2}right)right|^{2}r_{12}^{-1}
end{equation}
describes the average local potential at $mathbf{x}_{1}$ arising from the charge distribution from an electron in $chi_{j}$. And therefore the term $sum_{j neq i}left[int d mathbf{x}_{2}left|chi_{j}left(mathbf{x}_{2}right)right|^{2} r_{12}^{-1}right] chi_{i}left(mathbf{x}_{1}right)$ for an $N$-electron system gives the total averaged potential of $N-1$ electrons in other spin orbitals on the electron in $chi_{i}$.

However, if one expands the sum
begin{equation}
sum_{j neq i}left[int d mathbf{x}_{2}left|chi_{j}left(mathbf{x}_{2}right)right|^{2} r_{12}^{-1}right] chi_{i}left(mathbf{x}_{1}right) = left(int d mathbf{x}_{2}left|chi_{1}left(mathbf{x}_{2}right)right|^{2}r_{12}^{-1} + int d mathbf{x}_{2}left|chi_{2}left(mathbf{x}_{2}right)right|^{2}r_{12}^{-1} +…right) chi_{i}left(mathbf{x}_{1}right)
end{equation}
it is indeed a sum over the spin orbital but not of the electrons in the system. My interpretation for the sum is as follows: the total averaged potential acting on electron 1 due to the charge distribution of electron 2 in $N-1$ spin orbitals. I think it has something to do with the fact that electrons are indistinguishable so one cannot assign an electron to an orbital. However, I think the Hartree Fock integro-differential equation
begin{equation}
hleft(mathbf{x}_{1}right) chi_{i}left(mathbf{x}_{1}right)+sum_{j neq i}left[int d mathbf{x}_{2}left|chi_{j}left(mathbf{x}_{2}right)right|^{2} r_{12}^{-1}right] chi_{i}left(mathbf{x}_{1}right)-sum_{j neq i}left[int d mathbf{x}_{2} chi_{j}^{*}left(mathbf{x}_{2}right) chi_{i}left(mathbf{x}_{2}right) r_{12}^{-1}right] chi_{j}left(mathbf{x}_{1}right)=epsilon_{i} chi_{i}left(mathbf{x}_{1}right)
end{equation}
was derived for hydrogen molecule. Is it the same for an $N$-electron system?