Let $A$ be an $k$-affinoid algebra. Let $X=M(A)$ be the affinoid space associated to $A$ in the sense of Berkovich. There exists a natural morphism $\pi:X\to Y:=Spec(A)$ which sends $x$ to $ker (|\cdot|_x)$. In the book "Spectral theory and analytic geometry over non-archimedean fields" of Berkovich, he say that the functor $\mathcal{F}\mapsto\pi^*\mathcal{F}$ from the category of $\mathcal{O}_Y$-module to $\mathcal{O}_X$-module is exact and faithful. I don't know how to show that it is faithful.