This might need only a simple argument, but I simply cannot wrap my head around it.
Let $R$ be a commutative ring, $S$ a multiplicative subset with a distinguished element $c \in S$, and $A$ and $B$ be $R$-modules ($B$ can be assumed to be free and even of finite rank if this is necessary). Consider the diagram $\require{AMScd}$ \begin{CD} B @> \psi >> A @> \phi >> B \\ & @V V V @VV V\\ & & S^{-1} A @>> S^{-1} \phi > S^{-1} B \end{CD} with $S^{-1} \phi$ surjective (in fact $S^{-1} \psi \circ S^{-1} \phi$can be assumed surjective if necessary) and $\phi \circ \psi$ is multiplication by $c$.
I need to show that the cokernel of $\phi$ is annihilated by $c$. In other words that $$\text{coker} \, \phi \subset \text{ker} (\phi \circ \psi).$$
I have a hard time understanding how the surjectivity of the localized map relates to the problem.