We first construct the morphism $b : X' \to X$ by glueing the blowings up constructed over opens $V \subset X$ as in (5). By Constructions, Lemma \ref{constructions-lemma-relative-glueing} to do this it suffices to show that given $V \subset X$ open and two bounded complexes $\mathcal{E}^\bullet$ and $(\mathcal{E}')^\bullet$ of finite locally free $\mathcal{O}_V$-modules representing $E|_V$ the resulting blowing ups are canonically isomorphic. To do this, it suffices, by the universal property of blowing up of Divisors, Lemma \ref{divisors-lemma-universal-property-blowing-up}, to show that the ideals $\mathcal{J}_i$ and $\mathcal{J}'_i$ from Remark \ref{remark-complex-and-divisor-ideal} constructed using $\mathcal{E}^\bullet$ and $(\mathcal{E}')^\bullet$  locally differ by multiplication by an invertible ideal. We will in fact show that they differ locally by a power of the ideal sheaf $\mathcal{I}$ of $D$. By More on Algebra, Lemma \ref{more-algebra-lemma-compare-representatives-perfect} working locally it suffices to prove the relationship when $$ (\mathcal{E}')^\bullet = \mathcal{E}^\bullet \oplus ( \ldots \to 0 \to \mathcal{O}_V \xrightarrow{1} \mathcal{O}_V \to 0 \to \ldots) $$ with the two summands $\mathcal{O}_V$ placed in degrees $i$ and $i + 1$ say. Computing minors explicitly one finds that $\mathcal{J}'_{i + 1} = \mathcal{I}\mathcal{J}_{i + 1}$ and all other ideals stay the same. \medskip\noindent Thus we have the morphism $b : X' \to X$ agreeing locally with the blowing ups in (5). Of course this immediately gives us the effective Cartier divisor $D' = b^{-1}D$, its invertible ideal sheaf $\mathcal{I}'$ and the object $Q = L\eta_{\mathcal{I}'}Lb^*E$. See Remark \ref{remark-Leta} for the construction of $L\eta_{\mathcal{I}'}$. Since the construction commutes with restricting to opens we find that $Q|_{V'}$ is represented by the complex $\mathcal{Q}^\bullet$ over the open $V' = b^{-1}(V)$ constructed using $\mathcal{E}^\bullet$ over $V$. \medskip\noindent To finish the proof it suffices to show that the closed subschemes $T_V \subset V'$ constructed in Lemma \ref{lemma-complex-and-divisor-blowup-T} glue. Again by relative glueing, it suffices to show that the construction of $T$ does not depend on the choice of the complex $\mathcal{E}^\bullet$ representing $E|_V$. Again we reduce to the case where $$ (\mathcal{E}')^\bullet = \mathcal{E}^\bullet \oplus ( \ldots \to 0 \to \mathcal{O}_V \xrightarrow{1} \mathcal{O}_V \to 0 \to \ldots) $$ with the two summands $\mathcal{O}_V$ placed in degrees $i$ and $i + 1$ say. Note that in this case $(\mathcal{Q}')^\bullet$ and $\mathcal{Q}^\bullet$ differ as follows $$ (\mathcal{Q}')^\bullet = \mathcal{Q}^\bullet \oplus ( \ldots \to 0 \to (\mathcal{I}')^{i + 1}|_{V'} \xrightarrow{1} (\mathcal{I}')^{i + 1}|_{V'} \to 0 \to \ldots) $$ In the proof of Lemma \ref{lemma-complex-and-divisor-blowup-T} we defined $T \subset D'$ as the largest closed subscheme of $D'$ such that $\mathcal{Q}^i|_T$ is a direct sum of two parts compatible with the restriction to $T$ of the canonical split injective maps $$ c^i : \mathcal{Q}^i \longrightarrow (\mathcal{I}')^ib^*\mathcal{E}^i \oplus (\mathcal{I}')^{i + 1}b^*\mathcal{E}^{i + 1} $$ for all $i$. The direct sum decomposition for $(\mathcal{Q}')^\bullet$ in terms of $\mathcal{Q}^\bullet$ and the explicit complex $(\mathcal{I}')^{i + 1}|_{V'} \to (\mathcal{I}')^{i + 1}|_{V'}$ implies in a straightforward manner that $T$ plays the same role for $(\mathcal{Q}')^\bullet$ and the proof is complete.