# What is the duality between path cover and flow?

Let there be a bipartite directed graph $$G=(V,E)$$.

Let's say we have a path cover of the graph. In some texts it is said that this path cover "induces" a flow on $$G$$. What does this mean?

How can we extract a flow from a path cover (and similarly, extract a path cover from a flow)?

Presumably, the flow would send $$k$$ units of flow along the edge $$(u,v)$$, where $$k$$ is the number of paths in the path cover that traverse that edge.