The min cut capacity in a network based on a bipartite graph (Hall's Theorem)
The flow network $G'$ is constructed from the original bipartite graph $G=(L,R,E)$ by adding two new nodes $s,t$, connecting $s$ to all nodes in $L$, connecting all nodes in $R$ to $t$, and orienting all edges in $E$ from $L$ to $R$. And $s$-$t$ cut in $G'$ has the form $(S,\overline{S})$ where $s \in S$, $t \in \overline{S}$. The edges crossing the cut are:


Edges from $s$ to $\overline{S}$, that is, from $s$ to $\overline{S} \cap L$. There are $|L_2|$ of these.
Edges from $S$ to $t$, that is, from $S \cap R$ to $t$. There are $|R_1|$ of these.
Edges from $S \setminus \{s\}$ to $\overline{S} \setminus \{t\}$, that is, from $S \cap L$ to $\overline{S} \cap R$. There are $\mathrm{edges}(L_1,R_2)$ of these.


It is crucial that in $G'$ the edges are oriented, and that when counting edges crossing the cut $(S,\overline{S})$, we only count edges going from $S$ to $\overline{S}$.