Why does the charge goes down the line? Suppose we have two neutral charges connected by a conductor line. Now, we bring together two spheres with oppositely charges, and approximate it to our original system.

Now, imagine that we connect C and D by another line/wire, what exactly will happens?
I tried to answer this question as follow:
Since A and B have the same potential, we can call it $\phi$. Now, The potential of C near the sphere A is $\phi c>\phi$, and the D potential near the B sphere is $\phi d < \phi$, so that we have $\phi c > \phi d$. But, since we are dealing with conductors, $\phi c$ and $\phi d$ are the potentials throughout (constant through it) all the surface C and D, respectively.
When we connect C and D using a wire, since C has greater potential, the electric field in the wire will be directed down, from C to D. In Such way that, since electrons flow in the opposite direction of the electric field, they will rise up through the wire. So negative charge will flow from D to C.

But apparently, the answer is the opposite of the mine, the charge will goes down the wire. What is wrong with my attempt?
 A: You missed a very important detail while calculating the potential of the identical spheres C and D.
Initially they are neutral meaning they have equal number of oppo charged particles.
Lets consider of C for now (argument with D is exactly the same),
You bring the Sphere C close to A. What happens now? Charges in Sphere C will rearrange, but how? Such that some positive charges are closer to A. But since the C is neutral, the an equal amount of negative charges will move as well! Hence in the electrostatic condition, what you have is some positive charges on the surface towards A, while equal amount of opposite charges diametrically opposite. So then take the center of C and find the net potential there, because you have equal and opposite charges at diametrically opposite points, the net potential inside Sphere C is ZERO! (same for D)
Does it sound contradictory? Absolutely no.
Initially you had Sphere C at 0 Potential, now when you bring it closer to A the charges start rearraging, why does that happen? Since now Sphere A influences the Sphere C and so there is now a potential inside Sphere C, and the charges in C move so as to nullify this potential! Hence as we expect the end result would still have zero potential!
Now then if I connect C and D with a wire, what is the direction of current?
This situation is no longer ELECTROSTATICS because charges are moving.

The fact that Conductor has the same potential everywhere ONLY holds in electrostatics

Here you not supposed to look at the net Potential, instead look at the local Potential i.e. Potential just about the two points where you connect the wire to the two spheres. You will now find that the local potential near wire at C is NEGATIVE (because of accumulation of electrons) and using similar argument local potential at the wire for D is POSITIVE.
And since electrons flow from negative to positive i.e. from C to D.
