Why is augmenting path calculation O(m) I'm trying to learn augmenting paths and I got confused. I've found some other source that has the same setup that my slides have, therefore I will refer to that.
It is said that

Each augmenting path computation can be easily done in O(m) time.

in these lecture notes within a graph of m edges. I could not understand how it is O(m). Isn't trying to find an augmenting path that has capacity from source to the target an exhaustive process? Maybe I misunderstand what they mean as augmenting path computation but even if that is the case, the clarification of that would be helpful
As given in your lecture notes, and an augmenting path is an s-t path in the residual graph such that all edges on the path have a positive residual capacity.
There could be many augmenting paths in the graph. But you just need to find one such path. Therefore, you can apply DFS or BFS which takes $O(m)$ time to find one such path from $s$ to $t$.