Let $(\mathcal{G}_n)$ be the canonical extension as in Lemma \ref{lemma-canonically-algebraizable}. The grading on $A$ and $M_n$ determines an action $$ a : \mathbf{G}_m \times X \longrightarrow X $$ of the group scheme $\mathbf{G}_m$ on $X$ such that $(\widetilde{M_n})$ becomes an inverse system of $\mathbf{G}_m$-equivariant quasi-coherent $\mathcal{O}_X$-modules, see Groupoids, Example \ref{groupoids-example-Gm-on-affine}. Since $\mathfrak a$ and $I$ are homogeneous ideals the closed subschemes $Z$, $Y$ and the open subscheme $U$ are $\mathbf{G}_m$-invariant closed and open subschemes. The restriction $(\mathcal{F}_n)$ of $(\widetilde{M_n})$ is an inverse system of $\mathbf{G}_m$-equivariant coherent $\mathcal{O}_U$-modules. In other words, $(\mathcal{F}_n)$ is a $\mathbf{G}_m$-equivariant coherent formal module, in the sense that there is an isomorphism $$ \alpha : (a^*\mathcal{F}_n) \longrightarrow (p^*\mathcal{F}_n) $$ over $\mathbf{G}_m \times U$ satisfying a suitable cocycle condition. Since $a$ and $p$ are flat morphisms of affine schemes, by Lemma \ref{lemma-canonically-extend-base-change} we conclude that there exists a unique isomorphism $$ \beta : (a^*\mathcal{G}_n) \longrightarrow (p^*\mathcal{G}_n) $$ over $\mathbf{G}_m \times X$ restricting to $\alpha$ on $\mathbf{G}_m \times U$. The uniqueness guarantees that $\beta$ satisfies the corresponding cocycle condition. In this way each $\mathcal{G}_n$ becomes a $\mathbf{G}_m$-equivariant coherent $\mathcal{O}_X$-module in a manner compatible with transition maps. \medskip\noindent By Groupoids, Lemma \ref{groupoids-lemma-Gm-equivariant-module} we see that $\mathcal{G}_n$ with its $\mathbf{G}_m$-equivariant structure corresponds to a graded $A$-module $N_n$. The transition maps $N_{n + 1} \to N_n$ are graded module maps. Note that $N_n$ is a finite $A$-module and $N_n = N_{n + 1}/I^n N_{n + 1}$ because $(\mathcal{G}_n)$ is an object of $\textit{Coh}(X, I\mathcal{O}_X)$. Let $N$ be the finite graded $A$-module foud in Algebra, Lemma \ref{algebra-lemma-finiteness-graded}. Then $N_n = N/I^nN$, whence $(\mathcal{G}_n)$ is the completion of the coherent module associated to $N$, and a fortiori we see that (b) is true. \medskip\noindent To see (a) we have to unwind the situation described above a bit more. First, observe that the kernel and cokernel of $M_n \to H^0(U, \mathcal{F}_n)$ is $A_+$-power torsion (Local Cohomology, Lemma \ref{local-cohomology-lemma-finiteness-pushforwards-and-H1-local}). Observe that $H^0(U, \mathcal{F}_n)$ comes with a natural grading such that these maps and the transition maps of the system are graded $A$-module map; for example we can use that $(U \to X)_*\mathcal{F}_n$ is a $\mathbf{G}_m$-equivariant module on $X$ and use  Groupoids, Lemma \ref{groupoids-lemma-Gm-equivariant-module}. Next, recall that $(N_n)$ and $(H^0(U, \mathcal{F}_n))$ are pro-isomorphic by Definition \ref{definition-canonically-algebraizable} and Lemma \ref{lemma-canonically-algebraizable}. We omit the verification that the maps defining this pro-isomorphism are graded module maps. Thus $(N_n)$ and $(M_n)$ are pro-isomorphic in the category of graded $A$-modules modulo $A_+$-power torsion modules.