Fully faithfulness. Suppose that $(\mathcal{E}, n)$ and $(\mathcal{F}, m)$ are objects of $\colim_F \textit{Vect}(X)$. Let $(a, b) : G(\mathcal{E}, n) \to G(\mathcal{F}, m)$ be a morphism in the RHS. We may choose $N \gg 0$ and think of $a$ as a map $a : F^{N - n, *}\mathcal{E}|_Z \to F^{N - m, *}\mathcal{F}|_Z$ and $b$ as a map $b : F^{N - n, *}\mathcal{E}|_{X'} \to F^{N - m, *}\mathcal{F}|_{X'}$ agreeing over $E$. Choose a finite affine open covering $X = X_1 \cup \ldots \cup X_n$ such that $\mathcal{E}|_{X_i}$ and $\mathcal{F}|_{X_i}$ are finite free $\mathcal{O}_{X_i}$-modules. For each $i$ the base change $$ \xymatrix{ E_i \ar[r] \ar[d] & X'_i \ar[d] \\ Z_i \ar[r] & X_i } $$ is another almost blow up square as in Example \ref{example-two-generators}. For these squares we know that $$ \colim_F H^0(X_i, \mathcal{O}_{X_i}) = \colim_F H^0(Z_i, \mathcal{O}_{Z_i}) \times_{\colim_F H^0(E_i, \mathcal{O}_{E_i})} \colim_F H^0(X'_i, \mathcal{O}_{X'_i}) $$ by Lemma \ref{lemma-h-sheaf-colim-F} (see proof of the lemma). Hence after increasing $N$ we may assume the maps $a|_{Z_i}$ and $b|_{X'_i}$ come from maps $c_i : F^{N - n, *}\mathcal{E}|_{X_i} \to F^{N - m, *}\mathcal{F}|_{X_i}$. After possibly increasing $N$ we may assume $c_i$ and $c_j$ agree on $X_i \cap X_j$. Thus these maps glue to give the desired morphism $(\mathcal{E}, n) \to (\mathcal{F}, m)$ in the LHS. \medskip\noindent Essential surjectivity. Let $(\mathcal{F}, \mathcal{G}, \varphi)$ be a triple consisting of a finite locally free $\mathcal{O}_Z$-module $\mathcal{F}$, a finite locally free $\mathcal{O}_{X'}$-module $\mathcal{G}$, and an isomorphism $\varphi : \mathcal{F}|_E \to \mathcal{G}|_E$. We have to show that after replacing this triple by a Frobenius power pullback, it comes from a finite locally free $\mathcal{O}_X$-module. \medskip\noindent Noetherian reduction; we urge the reader to skip this paragraph. Recall that $X = \Spec(A)$ and $Z = \Spec(A/(f_1, f_2))$, $X' = \text{Proj}(A[T_0, T_1]/(f_2T_0 - f_1T_1))$, and $E = \mathbf{P}^1_Z$. By Limits, Lemma \ref{limits-lemma-descend-invertible-modules} we can find a finitely generated $\mathbf{F}_p$-subalgebra $A_0 \subset A$ containing $f_1$ and $f_2$ such that the triple $(\mathcal{F}, \mathcal{G}, \varphi)$ descends to $X_0 = \Spec(A_0)$ and $Z_0 = \Spec(A_0/(f_1, f_2))$, $X_0' = \text{Proj}(A_0[T_0, T_1]/(f_2T_0 - f_1T_1))$, and $E_0 = \mathbf{P}^1_{Z_0}$. Thus we may assume our schemes are Noetherian. \medskip\noindent Assume $X$ is Noetherian. We may choose a finite affine open covering $X = X_1 \cup \ldots \cup X_n$ such that $\mathcal{F}|_{Z \cap X_i}$ is free. Since we can glue objects of $\colim_F \textit{Vect}(X)$ in the Zariski topology (Lemma \ref{lemma-colim-F-Vect}), and since we already know fully faithfulness over $X_i$ and $X_i \cap X_j$ (see first paragraph of the proof), it suffices to prove the existence over each $X_i$. This reduces us to the case discussed in the next paragraph. \medskip\noindent Assume $X$ is Noetherian and $\mathcal{F} = \mathcal{O}_Z^{\oplus r}$. Using $\varphi$ we get an isomorphism $\mathcal{O}_E^{\oplus r} \to \mathcal{G}|_E$. Let $I = (f_1, f_2) \subset A$. Let $\mathcal{I} \subset \mathcal{O}_{X'}$ be the ideal sheaf of $E$; it is globally generated by $f_1$ and $f_2$. For any $n$ there is a surjection $$ (\mathcal{I}^n/\mathcal{I}^{n + 1})^{\oplus r} = \mathcal{I}^n/\mathcal{I}^{n + 1} \otimes_{\mathcal{O}_E} \mathcal{G}|_E \longrightarrow \mathcal{I}^n\mathcal{G}/\mathcal{I}^{n + 1}\mathcal{G} $$ Hence the first cohomology group of this module is zero. Here we use that $E = \mathbf{P}^1_Z$ and hence its structure sheaf and in fact any globally generated quasi-coherent module has vanishing $H^1$. Compare with More on Morphisms, Lemma \ref{more-morphisms-lemma-globally-generated-vanishing}. Then using the short exact sequences $$ 0 \to  \mathcal{I}^n\mathcal{G}/\mathcal{I}^{n + 1}\mathcal{G} \to \mathcal{G}/\mathcal{I}^{n + 1}\mathcal{G} \to \mathcal{G}/\mathcal{I}^n\mathcal{G} \to 0 $$ and induction, we see that $$ \lim H^0(X', \mathcal{G}/\mathcal{I}^n\mathcal{G}) \to H^0(E, \mathcal{G}|_E) = H^0(E, \mathcal{O}_E^{\oplus r}) = A/I^{\oplus r} $$ is surjective. By the theorem on formal functions (Cohomology of Schemes, Theorem \ref{coherent-theorem-formal-functions}) this implies that $$ H^0(X', \mathcal{G}) \to H^0(E, \mathcal{G}|_E) = H^0(E, \mathcal{O}_E^{\oplus r}) = A/I^{\oplus r} $$ is surjective. Thus we can choose a map $\alpha : \mathcal{O}_{X'}^{\oplus r} \to \mathcal{G}$ which is compatible with the given trivialization of $\mathcal{G}|_E$. Thus $\alpha$ is an isomorphism over an open neighbourhood of $E$ in $X'$. Thus every point of $Z$ has an affine open neighbourhood where we can solve the problem. Since $X' \setminus E \to X \setminus Z$ is an isomorphism, the same holds for points of $X$ not in $Z$. Thus another Zariski glueing argument finishes the proof.