One mark of being grown up is knowing when to let something go. That could mean dropping an argument, letting go of an expectation, or choosing to do something that needs to be done which neither of you wants to do.
The difficult part is fearing that letting go makes it look like you think she was right and you were wrong; and letting go after telling her you still think she is wrong is hardly grown up. This means that letting go in a mature way requires you to swallow your pride and do what is best for the relationship regardless of how you think it might make you look.
There are of course times when letting go is not the right choice. Some things matter enough that you need to work through them. The trick here is learning which things matter “that much”. I’ve come to realise that the list of what matters that much is rather small, especially when I eliminate the things that matter to me for reasons that are not reasonable, loving, or grown up.