On \(X=\CC\setminus\{0\}\), \(\exp\colon\OO_X\to\OO_X^\times\) is a surjection of sheaves, since \(\exp\) is surjective on every stalk, and it is not surjective on global sections: \(z\in\OO^\times(X)\) has no logarithm on \(X\).
The sheaf condition, the identity and gluing axioms, presheaves failing each, and sheafification.
Stalks and exactness, stalkwise exactness, and the class in \(H^1\) obstructing a global lift.