in Grothendieck's Tohoku paper (page 127), Grothendieck asserts that the category of holomorphic vector bundles $\mathbf{Bund}(X)$ over a fixed Riemann surface $X$ does not satisfy $\mathbf{AB2}$, but satisfies $\mathbf{AB1}$. Explicitly, he claims that $\mathbf{Bund}(X)$ has kernels and cokernels, however the canonical morphism $\overline{u}: \text{CoIm}(u) \longrightarrow \text{Im}(u)$ is not an isomorphism. At first I thought it was easy prove, but now I'm not so sure.

 I think it's useful to note that the category of vector bundles is not an abelian category in general (https://mathoverflow.net/questions/107101/is-the-category-of-vector-bundles-over-a-topological-space-abelian).

 Thanks in advance.