Sections of Cokernel sheaf

Question

Consider a sheaves morphism $f\colon \mathcal{F}\to \mathcal{G}$, where $\mathcal{F}$ and $\mathcal{G}$ are sheaves on a topological space $X$. Now we have the cokernel presheaf and its sheafification $\textit{Coker}(f)$. How can we describe the sections of this sheaf? In Griffith-Harris, Principles of Algebraic Geometry, on page 37, is stated that a section of $\textit{Coker}(f)$ on an open set $U\subseteq X$ is given by an open covering $(U_{\alpha})$ of $U$ and by a family $ (s_{\alpha})$ with $s_{\alpha}\in \mathcal{G}(U_{\alpha})$ such that for every overlapping open sets $ U_{\alpha}, U_{\beta}$ we have $ {s_{\alpha}}_{|U_{\alpha}\cap U_{\beta}}-{s_{\beta}}_{|U_{\alpha}\cap U_{\beta}}\in f_{U_{\alpha}\cap U_{\beta}}(\mathcal{F}(U_{\alpha}\cap U_{\beta}))$. My question is: why this is true? I know that a similar characterization holds for the sections of the sheafification of a separated presheaf, but the cokernel presheaf is not separated in general, if i am correct. For example this is true if $f$ is injective, but in general?