It's a paragraph in Vakil's FOAG, page 394, where the text is
14.2.6. The problem with locally free sheaves / vector bundles (as opposed to vector spaces).
Recall that $\mathscr{O}_{X}$-modules form an abelian category: we can talk about kernels, cokernels, and so forth, and we can do homological algebra. Similarly, vector spaces form an abelian category. But locally free sheaves (i.e., vector bundles), along with reasonably natural maps between them (those that arise as maps of $\mathscr{O}_{X}$-modules), don't form an abelian category. As a motivating example in the category of manifolds (cf. Warning 14.2.10), consider the map of the trivial line bundle on $\mathbb{R}$ (with coordinate $t$) to itself, corresponding to multiplying by the coordinate $t$. Then this map jumps rank, and if you try to define a kernel or cokernel you will get confused.
I can see "informally" that the map jumps rank: at $t=0$, the map has image a vector space of rank $0$ rather then $1$. But rigorously I cannot see where is the problem in this motivating example. Taking $\mathscr{O}_\Bbb{R}$ to be the sheaf of smooth functions. Then the trivial line bundle $\Bbb{R}$ is just $\mathscr{O}_\Bbb{R}$ as an $\mathscr{O}_\Bbb{R}$-module. And I want to calculate the kernel and cokernel for the map $\mathscr{O}_\Bbb{R} \to \mathscr{O}_\Bbb{R}$ defined by multiplying $t$. Then I get the kernel is just $0$, and the cokernel still is $\mathscr{O}_\Bbb{R}$. Both of them are still locally free sheaves(vector bundles). If so, then where is the "confused" point mentioned in the text? For this motivating example, it seems the kernel and cokernel still locally free sheaves? Or did I compute them wrong?
A related thread maybe Quasicoherent sheaves as smallest abelian category containing locally free sheaves. But after reading it, I still has no idea where is the problem in the quoted text.
Thank you in advance.