The trouble is that the wrongness is not really outside of me, it's a part of my mind, but the rest of my mindbelieves it is separate from it - disassociated from it. So really when I'm being right, I'm going into my persona and am rejecting my shadow, projecting it onto someone else and trying to position my persona-identified self as more correct, more truthful, more real.
This is a position of war, because always these situations of being right involve conflict - seemingly with someone else, but really with some part of myself. So long as someone else seems to be the opposite of my position, it becomes very easy to see my wrong-focused shadow projected onto them. It's a convenient way to get rid of my own wrongness and guilt.
So even if I seem very annoyed about someone being wrong, and I'm going to lengths to correct them or prove myself right, I secretly want them to be wrong so that I can put all of this sense of wrongfulness onto them, finding myself seemingly more innocent as a result. I get to look like the innocent victim who is just defending the truth, and they get to look like the evil monster who is hell-bent on destroying it.
This is all happening inside our own mind when we start to dig in our heels and be right. And the thing is, in this world there are many many situations in which, according to normal rules and laws and seeming "facts" (really illusions of truth), we can fall for this and believe that someone is "really wrong". We think there is a truth to the illusion that they are wrong, and react to it with righteousness in trying to separate ourselves from this perceived untruth. But separation itself is untruth.
We have to learn then to allow others to seem to be wrong, if they seem to be, recognizing it's not that they really are wrong, but only an illusion that they are, and that if we just react to that with ego we will just move into the persona and project our shadow guilt onto them. We need to let them be wrong, if they seem to be, and choose not to rise up in this mode of holier-than-thou, or the one who is good and righteous and correct. One person being wrong and one person being right is just an inner war, in which there is no peace.
Learning not to go into this mode of projection and wrong-making when, given the situation, it seems that you *actually* are correct, is a very subtle trap. It's easy to recognize you are being right when the other person is actually right and you're actually mistaken, on whatever ego grounds. But when the other person *seem* to be wrong, given the facts and so on, this makes it much harder to drop the 'role' of being the 'right one', or falling for the temptation of believing that something "really wrong" is happening outside of you.
Even when we are technically right, correct, or know a greater truth, we have to learn to surrender and not use it as a way to judge or to make others wrong for not seeing it. It's quite a subtle trap, and I fall for it all the time, lol. What ends up happening is I am defending the ego in me, thinking that I am defending some kind of truth, and only make myself be in conflict.
I can choose peace instead of this. I'd rather be happy than right.