We urge the reader to read the proof in the case where $A$ is generated in degree $1$ first, see Proposition \ref{proposition-coherent-modules-on-proj}. Let $f_1, \ldots, f_r \in A$ be homogeneous elements of positive degree which generate $A$ over $A_0$. Let $d$ be the lcm of the degrees $d_i$ of $f_i$. Let $M$ be a finite $A$-module. Let us show that $\widetilde{M}$ is zero if and only if $M$ is an irrelevant graded $A$-module (as defined above the statement of the proposition). Namely, let $x \in M$ be a homogeneous element. Choose $k \in \mathbf{Z}$ sufficiently small and let $N \to N'$ and $M \to N'$ be as in Lemma \ref{lemma-recover-tail-graded-module-general}. We may also pick $l$ sufficiently large such that $M_n \to N_n$ is an isomorphism for $n \geq l$. If $\widetilde{M}$ is zero, then $N = 0$. Thus for any $f \in A_+$ homogeneous with $\deg(f) + \deg(x) = nd$ and $nd > l$ we see that $fx$ is zero because $N_{nd} \to N'_{nd}$ and $M_{nd} \to N'_{nd}$ are isomorphisms. Hence $x$ is irrelevant. Conversely, assume $M$ is irrelevant. Then $M_{nd}$ is zero for $n \gg 0$ (see discussion above proposition). Clearly this implies that $M_{(f_i)} = M_{(f_i^{d/\deg(f_i)})} = 0$, whence $\widetilde{M} = 0$ by construction. \medskip\noindent It follows that the subcategory $\text{Mod}^{fg}_{A, irrelevant}$ is a Serre subcategory of $\text{Mod}^{fg}_A$ as the kernel of the exact functor $M \mapsto \widetilde M$, see Homology, Lemma \ref{homology-lemma-kernel-exact-functor} and Constructions, Lemma \ref{constructions-lemma-proj-sheaves}. Hence the quotient category on the left of the arrow is defined in Homology, Lemma \ref{homology-lemma-serre-subcategory-is-kernel}. To define the functor of the proposition, it suffices to show that the functor $M \mapsto \widetilde M$ sends irrelevant modules to $0$ which we have shown above. \medskip\noindent By Lemma \ref{lemma-coherent-on-proj-general} the proposed quasi-inverse makes sense. Namely, the lemma shows that $\mathcal{F} \longmapsto \bigoplus_{n \geq 0} \Gamma(X, \mathcal{F}(n))$ is a functor $\textit{Coh}(\mathcal{O}_X) \to \text{Mod}^{fg}_A$ which we can compose with the quotient functor $\text{Mod}^{fg}_A \to \text{Mod}^{fg}_A/\text{Mod}^{fg}_{A, irrelevant}$. \medskip\noindent By Lemma \ref{lemma-recover-tail-graded-module-general} the composite left to right to left is isomorphic to the identity functor. Namely, let $M$ be a finite graded $A$-module and let $k \in \mathbf{Z}$ sufficiently small and let $N \to N'$ and $M \to N'$ be as in Lemma \ref{lemma-recover-tail-graded-module-general}. Then the kernel and cokernel of $M \to N'$ are nonzero in only finitely many degrees, hence are irrelevant. Moreover, the kernel and cokernel of the map $N \to N'$ are zero in all sufficiently large degrees divisible by $d$, hence these are irrelevant modules too. Thus $M \to N'$ and $N \to N'$ are both isomorphisms in the quotient category, as desired. \medskip\noindent Finally, let $\mathcal{F}$ be a coherent $\mathcal{O}_X$-module. Set $M = \bigoplus_{n \in \mathbf{Z}} \Gamma(X, \mathcal{F}(n))$ viewed as a graded $A$-module, so that our functor sends $\mathcal{F}$ to $M_{\geq 0} = \bigoplus_{n \geq 0} M_n$. By Properties, Lemma \ref{properties-lemma-proj-quasi-coherent} the canonical map $\widetilde M \to \mathcal{F}$ is an isomorphism. Since the inclusion map $M_{\geq 0} \to M$ defines an isomorphism $\widetilde{M_{\geq 0}} \to \widetilde M$ we conclude that the composite right to left to right is isomorphic to the identity functor as well.