Let $B$ be a graded ring an $M$ be a $\mathbb Z$-graded $B$-module. We can associate to $M$ a sheaf of modules on $\operatorname{Proj} B$ by defining the sheaf on the principal open sets $D_+(f)$ to be $(M_{(f)})^\sim$ (where this is the associated module in the affine case) and checking that these agree on overlaps.
However, there is a second construction, which appears on page $165$ of Qing Liu's Algebraic Geometry and Arithmetic Curves. Consider the canonical injection $f:\operatorname{Proj B}\rightarrow \operatorname{Spec} B$. This is continuous. Consider the sheaf $\mathcal F = M^\sim$ on $\operatorname{Spec} B$ (again, this is the associated sheaf in the affine case). We can take the subsheaf of $f^{-1}\mathcal F$ consisting of all degree 0 elements to be the sheaf associated to $M$ on $\operatorname{Proj} B$.
(As @Hanno notes in the comments, there is an issue here. There doesn't seem to be a way to turn this into a sheaf of modules, because $f$ is a merely a continuous map, not a morphism. I would also appreciate comments on this issue.)
I would like to know why these constructions agree. I know that $(f^{-1} \mathcal F)_{\mathfrak p} = M_\mathfrak p$, so taking the homogeneous elements, we see that the two sheafs have the same stalks. But I am rather uncomfortable with the inverse image construction, and do not see how to construct an isomorphism.