While proving the equivalence of Sheaf cohomology (defined by using injective resolutions), Tennison (in his book 'Sheaf Theory' on page 146) says :
If we let $S$ be a presheaf such that the sequence $ 0 \to R' \to R \to S \to 0 $ is exact in Presh/X, we see that $S$ has as sheafification the zero sheaf and there is an exact sequence < long exact sequence corresponding to the above short exxact sequence>.
Here $R' \to R$ is the sheafification map of the presheaf $R'$. I can understand that $S$ can be shosen to be the Presheaf cokernel of the sheafification map. My question is : But how do we know that this map is exact at $R'$ ? Edit : As pointed out in the comment, this need not be exact at $R'$. So the question is : How do we tweak the proof in the book so that it becomes valid ?