Unique min-cut means unique max flow?
I do not agree with the downvoters of your question. This questions seems like a legit misunderstanding of the statement of the min-cost-max-flow theorem.
The theorem states that the value of the sum of edge weights in a minimum $s$-$t$ cut of a graph is equal to the value of the maximum $s$-$t$ flow in the graph interpreted as a flow network. It does not establish a one-to-one correspondence between flows and cuts.
In fact, this correspondence cannot be established in general. Here is a counter example of an unweighted undirected graph (all weights/capacities are unit):

$1$-$5$ max-flow/min-cut is equal to one. While there is a single minimum cut ({1 - 2}). There are a lot of possible maximum flows (take the unit of flow entering 2 and distribute it between 3 and 4 as you like).