A locally free sheaf $\mathcal E$ has a sheaf fibre $\mathcal E_x$ at $x$ but also a vector fibre $\mathcal E[x]=\mathcal E_x \otimes _ {\mathcal O_x} k(x)$. The fact that tensoring is not exact explains the paradox that a locally free subsheaf of a locally free sheaf does not yield a sub-vector bundle of a vector bundle in the above equivalence. The contrasting notation $\mathcal E[x]$ versus $\mathcal E_x$ (that I learned from German mathematicians) may help clarify this subtle point .