What will be the form of coulomb and exchange operators for electron m in an n-electron molecule?

I know that the Fock operator for electron $m$ in $n$-electron molecule would be:


$$\hat{f}\!(m)u\_i(m) = \varepsilon\_iu\_i(m)$$


$$\hat{f}\!(m) \equiv -\frac 1 2 ∇\_m^2 - \sum\_α\frac{Z\_α}{r\_{ma}} + \sum\_{j = 1}^n\left[\hat{j}\_j(m) - \hat{k}\_j(m)\right]$$


And the Coulomb operator according to [Wikipedia](https://en.wikipedia.org/wiki/Coulomb_operator) is given by:


$$\hat{J}\_{\!j}(1)f(1) = f(1)\int\left|\varphi\_j(2)\right|^2\frac{1}{r\_{12}}\mathrm dr\_2$$


My understanding of this equation is that we are finding the interaction of $f(1)$ with the electrons of $j$-th orbital (we take interaction with electron 2 and twice that result, hence $2J$ is used in the expression for Fock operator), and $r\_{12}$ is the separation between our electron in $f(1)$ and 2nd electron in $j$-th orbital.


This is the expression for exchange operator for electron $1$ in $j$-th orbital with an arbitrary function '$f$'. What would be the expression of Coulomb and exchange operators for electron $m$? Will we consider the $1$st electron of $(m/2)$-th orbital? Or will we simply put $m$ in place of $1$?


Or have I understood the whole thing incorrectly?


