I know that a max flow problem for undirected graphs has a standard reduction to directed graphs in order to apply Ford–Fulkerson algorithm and getting a feasible solution.
Here's my doubt. An undirected edge $(u,v)$ with capacity $1$, will correspond to two directed edges $(u,v), (v,u)$ both with capacity $1$. However, the solution cannot use both edges since I would lose the correspondence with the original problem.
What am I missing?