When following the 連用形（れんようけい） (conjunctive form) of another verb, the う - Verb 合（あ）う
means to do that action in concert with another person/group. This is often translated as ‘to mutually do (A)’, or ‘to do (A) for one another’. Realistically, it just means ‘to do (A) reciprocally’, with involvement from both sides being required. In this usage, 合（あ）う
is considered to be a 接尾後（せつびご） (suffix).
This construction is often used with expressions like お互（たが）いに ‘mutually’, which simply emphasizes the point.
I am going to talk with my girlfriend about our future.
The kids were fighting with each other over a toy, and that reminded me of my older brother.
My friend and I showed each other our Pokemon cards.
is regularly translated as ‘to fit’, or ‘to meet’ in English, its meaning is somewhat broader in Japanese. 合（あ）う
means to perform an action ‘as one’, where it is not possible (or would be difficult) to complete the action by oneself. Due to this, it is regularly used to describe ideal situations in which individuals should be doing something together, without thinking about their own personal gain.
At times like this, it is important to help each other out.
We are husband and wife, so no matter what happens, we need to cooperate.
may also be used to in regard to sports/contests. In these cases, it may seem like players are not doing the action ‘for’ each other, but involvement from both sides is required to make the act possible.