Let $Y' \subset X'$ be the exceptional divisor. Let $Z' \subset Y'$ be the inverse image of $Z \subset Y$. Then $U' = X' \setminus Z'$ is the inverse image of $U$. With $\delta^{Y'}_{Z'}$ as in (\ref{equation-delta-Z}) we set $$ T' = \{y' \in Y' \mid \delta^{Y'}_{Z'}(y') = 1\text{ or }2\} \subset T = \{y' \in Y' \mid \delta^{Y'}_{Z'}(y') = 1\} $$ These are specialization stable subsets of $U' \cap Y' = Y' \setminus Z'$. Consider the object $(b|_{U'}^*\mathcal{F}_n)$ of $\textit{Coh}(U', I\mathcal{O}_{U'})$, see Cohomology of Schemes, Lemma \ref{coherent-lemma-inverse-systems-pullback}. For $y' \in U' \cap Y'$ let us denote $$ \mathcal{F}_{y'}^\wedge = \lim (b|_{U'}^*\mathcal{F}_n)_{y'} $$ the ``stalk'' of this pullback at $y'$. We claim that conditions (a), (b), (c), (d), and (e) of Lemma \ref{lemma-improvement-formal-coherent-module-better} hold for the object $(b|_{U'}^*\mathcal{F}_n)$ on $U'$ with $d$ replaced by $1$ and the subsets $T' \subset T \subset U' \cap Y'$. Condition (a) holds because $Y'$ is an effective Cartier divisor and hence locally cut out by $1$ equation. Condition (e) holds by Lemma \ref{lemma-discussion} parts (1) and (2). To prove (b), (c), and (d) we need some preparation. \medskip\noindent Let $y' \in U' \cap Y'$ and let $\mathfrak p' \subset \mathcal{O}_{X', y'}^\wedge$ be a prime ideal not contained in $V(I\mathcal{O}_{X', y'}^\wedge)$. Denote $y = b(y') \in U \cap Y$. Choose $f \in I$ such that $y'$ is contained in the spectrum of the affine blowup algebra $A[\frac{I}{f}]$, see Divisors, Lemma \ref{divisors-lemma-blowing-up-affine}. For any $A$-algebra $B$ denote $B' = B[\frac{IB}{f}]$ the corresponding affine blowup algebra. Denote $I$-adic completion by ${\ }^\wedge$. By our choice of $f$ we get a ring map $(\mathcal{O}_{X, y}^\wedge)' \to \mathcal{O}_{X', y'}^\wedge$. If we let $\mathfrak q' \subset (\mathcal{O}_{X, y}^\wedge)'$ be the inverse image of $\mathfrak m_{y'}^\wedge$, then we see that $((\mathcal{O}_{X, y}^\wedge)'_{\mathfrak q'})^\wedge = \mathcal{O}_{X', y'}^\wedge$. Let $\mathfrak p \subset \mathcal{O}_{X, y}^\wedge$ be the corresponding prime. At this point we have a commutative diagram $$ \xymatrix{ \mathcal{O}_{X, y}^\wedge \ar[d] \ar[r] & (\mathcal{O}_{X, y}^\wedge)' \ar[d]_\alpha \ar[r] & (\mathcal{O}_{X, y}^\wedge)'_{\mathfrak q'} \ar[d] \ar[r]_\beta & \mathcal{O}_{X', y'}^\wedge \ar[d] \\ \mathcal{O}_{X, y}^\wedge/\mathfrak p \ar[r] & (\mathcal{O}_{X, y}^\wedge/\mathfrak p)' \ar[r] & (\mathcal{O}_{X, y}^\wedge/\mathfrak p)'_{\mathfrak q'} \ar[r]^\gamma & ((\mathcal{O}_{X, y}^\wedge/\mathfrak p)'_{\mathfrak q'})^\wedge \ar[d] \\  & & & \mathcal{O}_{X', y'}^\wedge/\mathfrak p' } $$ whose vertical arrows are surjective. By More on Algebra, Lemma \ref{more-algebra-lemma-completion-dimension} and the dimension formula (Algebra, Lemma \ref{algebra-lemma-dimension-formula}) we have $$ \dim(((\mathcal{O}_{X, y}^\wedge/\mathfrak p)'_{\mathfrak q'})^\wedge) = \dim((\mathcal{O}_{X, y}^\wedge/\mathfrak p)'_{\mathfrak q'}) = \dim(\mathcal{O}_{X, y}^\wedge/\mathfrak p) - \text{trdeg}(\kappa(y')/\kappa(y)) $$ Tracing through the definitions of pullbacks, stalks, localizations, and completions we find $$ (\mathcal{F}_y^\wedge)_{\mathfrak p} \otimes_{(\mathcal{O}_{X, y}^\wedge)_\mathfrak p} (\mathcal{O}_{X', y'}^\wedge)_{\mathfrak p'} = (\mathcal{F}_{y'}^\wedge)_{\mathfrak p'} $$ Details omitted. The ring maps $\beta$ and $\gamma$ in the diagram are flat with Gorenstein (hence Cohen-Macaulay) fibres, as these are completions of rings having a dualizing complex. See Dualizing Complexes, Lemmas \ref{dualizing-lemma-formal-fibres-gorenstein} and \ref{dualizing-lemma-dualizing-gorenstein-formal-fibres} and the discussion in More on Algebra, Section \ref{more-algebra-section-properties-formal-fibres}. Observe that $(\mathcal{O}_{X, y}^\wedge)_\mathfrak p = (\mathcal{O}_{X, y}^\wedge)'_{\tilde{\mathfrak p}}$ where $\tilde{\mathfrak p}$ is the kernel of $\alpha$ in the diagram. On the other hand, $(\mathcal{O}_{X, y}^\wedge)'_{\tilde{\mathfrak p}} \to (\mathcal{O}_{X', y'}^\wedge)_{\mathfrak p'}$ is flat with CM fibres by the above. Whence $(\mathcal{O}_{X, y}^\wedge)_\mathfrak p  \to (\mathcal{O}_{X', y'}^\wedge)_{\mathfrak p'}$ is flat with CM fibres. Using Algebra, Lemma \ref{algebra-lemma-apply-grothendieck-module} we see that $$ \text{depth}((\mathcal{F}_{y'}^\wedge)_{\mathfrak p'}) = \text{depth}((\mathcal{F}_y^\wedge)_{\mathfrak p}) + \dim(F_\mathfrak r) $$ where $F$ is the generic formal fibre of $(\mathcal{O}_{X, y}^\wedge/\mathfrak p)'_{\mathfrak q'}$ and $\mathfrak r$ is the prime corresponding to $\mathfrak p'$. Since $(\mathcal{O}_{X, y}^\wedge/\mathfrak p)'_{\mathfrak q'}$ is a universally catenary local domain, its $I$-adic completion is equidimensional and (universally) catenary by Ratliff's theorem (More on Algebra, Proposition \ref{more-algebra-proposition-ratliff}). It then follows that $$ \dim(((\mathcal{O}_{X, y}^\wedge/\mathfrak p)'_{\mathfrak q'})^\wedge) = \dim(F_\mathfrak r) + \dim(\mathcal{O}_{X', y'}^\wedge/\mathfrak p') $$ Combined with Lemma \ref{lemma-change-distance-function} we get \begin{equation} \label{equation-one} \begin{aligned} & \text{depth}((\mathcal{F}_{y'}^\wedge)_{\mathfrak p'}) + \delta^{Y'}_{Z'}(y') \\ & = \text{depth}((\mathcal{F}_y^\wedge)_{\mathfrak p}) + \dim(F_\mathfrak r) + \delta^{Y'}_{Z'}(y') \\ & \geq \text{depth}((\mathcal{F}_y^\wedge)_{\mathfrak p}) + \delta^Y_Z(y) + \dim(F_\mathfrak r) + \text{trdeg}(\kappa(y')/\kappa(y)) - (d - 1) \\ & = \text{depth}((\mathcal{F}_y^\wedge)_{\mathfrak p}) + \delta^Y_Z(y) - (d - 1) + \dim(\mathcal{O}_{X, y}^\wedge/\mathfrak p) - \dim(\mathcal{O}_{X', y'}^\wedge/\mathfrak p') \end{aligned} \end{equation} Please keep in mind that $\dim(\mathcal{O}_{X, y}^\wedge/\mathfrak p) \geq \dim(\mathcal{O}_{X', y'}^\wedge/\mathfrak p')$. Rewriting this we get \begin{equation} \label{equation-two} \begin{aligned} & \text{depth}((\mathcal{F}_{y'}^\wedge)_{\mathfrak p'}) + \dim(\mathcal{O}_{X', y'}^\wedge/\mathfrak p') + \delta^{Y'}_{Z'}(y') \\ & \geq \text{depth}((\mathcal{F}_y^\wedge)_{\mathfrak p}) + \dim(\mathcal{O}_{X, y}^\wedge/\mathfrak p) + \delta^Y_Z(y) - (d - 1) \end{aligned} \end{equation} This inequality will allow us to check the remaning conditions. \medskip\noindent Conditions (b) and (d) of Lemma \ref{lemma-improvement-formal-coherent-module-better}. Assume $V(\mathfrak p') \cap V(I\mathcal{O}_{X', y'}^\wedge) = \{\mathfrak m_{y'}^\wedge\}$. This implies that $\dim(\mathcal{O}_{X', y'}^\wedge/\mathfrak p') = 1$ because $Z'$ is an effective Cartier divisor. The combination of (b) and (d) is equivalent with $$ \text{depth}((\mathcal{F}_{y'}^\wedge)_{\mathfrak p'}) + \delta^{Y'}_{Z'}(y') > 2 $$ If $(\mathcal{F}_n)$ satisfies the inequalities in (3)(b) then we immediately conclude this is true by applying (\ref{equation-two}). If $(\mathcal{F}_n)$ satisfies (3)(a), i.e., the $(d + 1, d + 2)$-inequalities, then we see that in any case $$ \text{depth}((\mathcal{F}_y^\wedge)_{\mathfrak p}) + \delta^Y_Z(y) \geq d + 1 \quad\text{or}\quad \text{depth}((\mathcal{F}_y^\wedge)_{\mathfrak p}) + \dim(\mathcal{O}_{X, y}^\wedge/\mathfrak p) + \delta^Y_Z(y) > d + 2 $$ Looking at (\ref{equation-one}) and (\ref{equation-two}) above this gives what we want except possibly if $\dim(\mathcal{O}_{X, y}^\wedge/\mathfrak p) = 1$. However, if $\dim(\mathcal{O}_{X, y}^\wedge/\mathfrak p) = 1$, then we have $V(\mathfrak p) \cap V(I\mathcal{O}_{X, y}^\wedge) = \{\mathfrak m_y^\wedge\}$ and we see that actually $$ \text{depth}((\mathcal{F}_y^\wedge)_{\mathfrak p}) + \delta^Y_Z(y) > d + 1 $$ as $(\mathcal{F}_n)$ satisfies the $(d + 1, d + 2)$-inequalities and we conclude again. \medskip\noindent Condition (c) of Lemma \ref{lemma-improvement-formal-coherent-module-better}. Assume $V(\mathfrak p') \cap V(I\mathcal{O}_{X', y'}^\wedge) \not = \{\mathfrak m_{y'}^\wedge\}$. Then condition (c) is equivalent to $$ \text{depth}((\mathcal{F}_{y'}^\wedge)_{\mathfrak p'}) + \delta^{Y'}_{Z'}(y') \geq 2 \quad\text{or}\quad \text{depth}((\mathcal{F}_{y'}^\wedge)_{\mathfrak p'}) + \dim(\mathcal{O}_{X', y'}^\wedge/\mathfrak p') + \delta^{Y'}_{Z'}(y') > 3 $$ If $(\mathcal{F}_n)$ satisfies the inequalities in (3)(b) then we see the second of the two displayed inequalities holds true by applying (\ref{equation-two}). If $(\mathcal{F}_n)$ satisfies (3)(a), i.e., the $(d + 1, d + 2)$-inequalities, then this follows immediately from (\ref{equation-one}) and (\ref{equation-two}). This finishes the proof of our claim. \medskip\noindent Choose $(b|_{U'}^*\mathcal{F}_n) \to (\mathcal{F}_n'')$ and $(\mathcal{H}_n)$ in $\textit{Coh}(U', I\mathcal{O}_{U'})$ as in Lemma \ref{lemma-improvement-formal-coherent-module-better}. For any affine open $W \subset X'$ observe that $\delta^{W \cap Y'}_{W \cap Z'}(y') \geq \delta^{Y'}_{Z'}(y')$ by Lemma \ref{lemma-discussion} part (7). Hence we see that $(\mathcal{H}_n|_W)$ satisfies the assumptions of Lemma \ref{lemma-cd-1-canonical}. Thus $(\mathcal{H}_n|_W)$ extends canonically to $W$. Let $(\mathcal{G}_{W, n})$ in $\textit{Coh}(W, I\mathcal{O}_W)$ be the canonical extension as in Lemma \ref{lemma-canonically-algebraizable}. By Lemma \ref{lemma-canonically-extend-base-change} we see that for $W' \subset W$ there is a unique isomorphism $$ (\mathcal{G}_{W, n}|_{W'}) \longrightarrow (\mathcal{G}_{W', n}) $$ compatible with the given isomorphisms $(\mathcal{G}_{W, n}|_{W \cap U}) \cong (\mathcal{H}_n|_{W \cap U})$. We conclude that there exists an object $(\mathcal{G}_n)$ of $\textit{Coh}(X', I\mathcal{O}_{X'})$ whose restriction to $U$ is isomorphic to $(\mathcal{H}_n)$. \medskip\noindent If $A$ is $I$-adically complete we can finish the proof as follows. By Grothedieck's existence theorem (Cohomology of Schemes, Lemma \ref{coherent-lemma-existence-projective}) we see that $(\mathcal{G}_n)$ is the completion of a coherent $\mathcal{O}_{X'}$-module. Then by Cohomology of Schemes, Lemma \ref{coherent-lemma-existence-easy} we see that $(b|_{U'}^*\mathcal{F}_n)$ is the completion of a coherent $\mathcal{O}_{U'}$-module $\mathcal{F}'$. By Cohomology of Schemes, Lemma \ref{coherent-lemma-inverse-systems-push-pull} we see that there is a map $$ (\mathcal{F}_n) \longrightarrow ((b|_{U'})_*\mathcal{F}')^\wedge $$ whose kernel and cokernel is annihilated by a power of $I$. Then finally, we win by applying Lemma \ref{lemma-map-kernel-cokernel-on-closed}. \medskip\noindent If $A$ is not complete, then, before starting the proof, we may replace $A$ by its completion, see Lemma \ref{lemma-algebraizable}. After completion the assumptions still hold: this is immediate for condition (3), follows from Dualizing Complexes, Lemma \ref{dualizing-lemma-ubiquity-dualizing} for condition (1), and from Divisors, Lemma \ref{divisors-lemma-flat-base-change-blowing-up} for condition (2). Thus the complete case implies the general case.