Flows in graphs and the homology of free categories

We study the $R$-module of generalized flows in a graph with coefficients in the $R$-representation of the graph over a ring $R$ with 1 and show that this $R$-module is isomorphic to the first derived functor of the colimit. We generalize Kirchhoff's laws and build an exact sequence for calculating the $R$-module of flows in the union of graphs.