Let \(\varphi : \mcf \to \mcg\) be a morphism of sheaves.
The kernel is \(U \mapsto \ker \varphi(U)\), which is already a sheaf.
The image is the sheafification of \(U \mapsto \im \varphi(U)\).
The cokernel is the sheafification of \(U \mapsto \mcg(U)/\im \varphi(U)\).
A section of \(\mcf\) lies in \(\ker\varphi(U)\) if and only if its restrictions to the members of an open cover of \(U\) do, so the kernel presheaf satisfies both sheaf axioms. A section \(t\in\mcg(U)\) lies in \(\im\varphi(U)\) when there exists \(s\in\mcf(U)\) with \(\varphi(s)=t\); local preimages on an open cover of \(U\) need not agree on overlaps, so they need not glue to a preimage on \(U\).
The image presheaf is a subpresheaf of the sheaf \(\mcg\), so it satisfies the identity axiom; it can fail gluing, and sheafification adds the sections of \(\mcg\) that lie in the image locally. The cokernel presheaf can fail both axioms; sheafification sends to \(0\) the sections that are locally zero.
(The image presheaf of the exponential map)
On \(X=\CC\sm\ts{0}\), let \(\varphi=\exp\colon\OO_X\to\OO_X^\times\). The section \(z\in\OO_X^\times(X)\) has a logarithm on every simply connected open subset of \(X\) but none on \(X\). So \(z\) is not in the image presheaf on \(X\), and its class in the cokernel presheaf on \(X\) is nonzero but locally zero. The image sheaf is \(\OO_X^\times\) and the cokernel sheaf is \(0\).