Show a reduction of max flow to min cost (not min cost max flow!!)
Consider a graph $G$ and let $s$ (resp. $t$) be the source (resp. target) vertex for your max-flow problem. Let $U$ be an upper bound to the max flow (for example the sum of the capacities of the edges leaving $s$, or the sum of the capacities of the edges entering $t$).
Construct an instance of min-cost flow where you want to send $U$ units of flow from $s$ to $t$ and the graph $G$ is modified as follows:
- All edges of $G$ have cost $0$
- Add a new edge $(s,t)$ with cost $1$ and capacity $U$.
If $x$ is the cost of a min-cost flow on this graph, then the max flow in $G$ is $U-x$.