I am readying Hatcher page 178 where he tries to give the idea of cohomology in the case where our space $X$ is a graph. Now in the 4th paragraph from the top he says this:
The cohomology group $H^1(X,G) = \Delta^1(X;G)/\textrm{Im} \delta$ will be trivial iff the equation $\delta \varphi = \psi$ has a solution $\varphi \in \Delta^0 (X;G)$ for each $\psi \in \Delta^1 (X;G)$. Solving this equation means deciding whether specifying the change in $\varphi$ across each edge of $X$ determines an actual function $\varphi \in \Delta^0(X;G)$.
What does he mean by that last sentence in bold in the paragraph above?
Thanks.