On \(X = \CC^*\) take the exponential sequence \[ 0 \to \ul{\ZZ} \xrightarrow{\ 2\pi i\ } \OO_X \xrightarrow{\ \exp\ } \OO_X^* \to 0 . \] It is exact as a sequence of sheaves, because it is exact on stalks: near any point a nonvanishing holomorphic function has a logarithm.
It is not exact as a sequence of presheaves. Cover \(\CC^*\) by contractible opens \(U_i\), on each of which \(\log\) exists, so \(\id_{U_i} \in \OO^*(U_i)\) lies in the image presheaf. These sections agree on overlaps and glue to \(\id_{\CC^*}\), which is not in the image presheaf: a global logarithm on \(\CC^*\) does not exist.
So the image presheaf fails the gluing axiom, and sheafifying it is exactly what supplies the missing section.
The same sequence, taken in cohomology, is where the Picard group comes from: \[ H^1(X, \OO) \to H^1(X, \OO^*) \to H^2(X, \ZZ) , \] with \(H^1(X,\OO^*) = \Pic(X)\) and the right-hand map the first Chern class. The failure to glue here and the failure of surjectivity on global sections in PR-C9ZEK are the same phenomenon counted twice.
Dependencies
D-A7LCTKernel, image and cokernel sheaves, and which need sheafifyinguses