We first describe the maps $r'$ and $\theta'$. Observe that $\mathcal{L}'$ is $f'$-ample, see Lemma \ref{lemma-thicken-property-relatively-ample}. There is a canonical map of quasi-coherent graded $\mathcal{O}_{S'}$-algebras $\mathcal{A}' \to \bigoplus_{d \geq 0} (f')_*(\mathcal{L}')^{\otimes d}$ which is an isomorphism in degrees $\geq d_0$. Hence this induces an isomorphism on relative Proj compatible with the Serre twists of the structure sheaf, see Constructions, Lemma \ref{constructions-lemma-eventual-iso-graded-rings-map-relative-proj}. Hence we get the morphism $r'$ by Morphisms, Lemma \ref{morphisms-lemma-characterize-relatively-ample} (which in turn appeals to the construction given in Constructions, Lemma \ref{constructions-lemma-invertible-map-into-relative-proj}) and it is an isomorphism by Morphisms, Lemma \ref{morphisms-lemma-proper-ample-is-proj}. We get the map $\theta'$ from Constructions, Lemma \ref{constructions-lemma-invertible-map-into-relative-proj}. By Properties, Lemma \ref{properties-lemma-ample-gcd-is-one} we find that $\theta'$ is an isomorphism (this also uses that the morphism $r'$ over affine opens of $S'$ is the same as the morphism from Properties, Lemma \ref{properties-lemma-map-into-proj} as is explained in the proof of Morphisms, Lemma \ref{morphisms-lemma-proper-ample-is-proj}). \medskip\noindent Assuming the vanishing and local freeness stated in parts (1) and (2), the functoriality of the construction can be seen as follows. Suppose that $h : T \to S'$ is a morphism of schemes, denote $f_T : X'_T \to T$ the base change of $f'$ and $\mathcal{L}_T$ the pullback of $\mathcal{L}$ to $X'_T$. By cohomology and base change (as formulated in Derived Categories of Schemes, Lemma \ref{perfect-lemma-compare-base-change} for example) we have the corresponding vanishing over $T$ and moreover $h^*\mathcal{A}'_d = f_{T, *}\mathcal{L}_T^{\otimes d}$ (and thus the local freeness of pushforwards as well as the finite generation of the corresponding graded $\mathcal{O}_T$-algebra $\mathcal{A}_T$). Hence the morphism $r_T : X_T \to \underline{\text{Proj}}_T(\bigoplus f_{T, *}\mathcal{L}_T^{\otimes d})$ is simply the base change of $r'$ to $T$ and the pullback of $\theta'$ is the map $\theta_T$. \medskip\noindent Having said all of the above, we see that it suffices to prove (1), (2), and (3). Pick $d_0$ such that $R^pf_*\mathcal{L}^{\otimes d} = 0$ for all $d \geq d_0$ and $p > 0$, see Cohomology of Schemes, Lemma \ref{coherent-lemma-coherent-proper-ample}. We claim that $d_0$ works. \medskip\noindent By cohomology and base change (Derived Categories of Schemes, Lemma \ref{perfect-lemma-flat-proper-perfect-direct-image-general}) we see that $E'_d = Rf'_*(\mathcal{L}')^{\otimes d}$ is a perfect object of $D(\mathcal{O}_{S'})$ and its formation commutes with arbitrary base change. In particular, $E_d = Li^*E'_d = Rf_*\mathcal{L}^{\otimes d}$. By Derived Categories of Schemes, Lemma \ref{perfect-lemma-vanishing-implies-locally-free} we see that for $d \geq d_0$ the complex $E_d$ is isomorphic to the finite locally free $\mathcal{O}_S$-module $f_*\mathcal{L}^{\otimes d}$ placed in cohomological degree $0$. Then by Derived Categories of Schemes, Lemma \ref{perfect-lemma-open-where-cohomology-in-degree-i-rank-r} we conclude that $E'_d$ is isomorphic to a finite locally free module placed in cohomological degree $0$. Of course this means that $E'_d = \mathcal{A}'_d[0]$, that $R^pf'_*(\mathcal{L}')^{\otimes d} = 0$ for $p > 0$, and that $\mathcal{A}'_d$ is finite locally free. This proves (1) and (2). \medskip\noindent The last thing we have to show is finite presentation of $\mathcal{A}'$ as a sheaf of $\mathcal{O}_{S'}$-algebras (this notion was introduced in Properties, Section \ref{properties-section-extending-quasi-coherent-sheaves}). Let $U' = \Spec(R') \subset S'$ be an affine open. Then $A' = \mathcal{A}'(U')$ is a graded $R'$-algebra whose graded parts are finite projective $R'$-modules. We have to show that $A'$ is a finitely presented $R'$-algebra. We will prove this by reduction to the Noetherian case. Namely, we can find a finite type $\mathbf{Z}$-subalgebra $R'_0 \subset R'$ and a pair\footnote{With the same properties as those enjoyed by $X' \to S'$ and $\mathcal{L}'$, i.e., $X'_0 \to \Spec(R'_0)$ is flat and proper and $\mathcal{L}'_0$ is ample.} $(X'_0, \mathcal{L}'_0)$ over $R'_0$ whose base change is $(X'_{U'}, \mathcal{L}'|_{X'_{U'}})$, see Limits, Lemmas \ref{limits-lemma-descend-modules-finite-presentation}, \ref{limits-lemma-descend-invertible-modules}, \ref{limits-lemma-eventually-proper}, \ref{limits-lemma-descend-flat-finite-presentation}, and \ref{limits-lemma-limit-ample}. Cohomology of Schemes, Lemma \ref{coherent-lemma-coherent-proper-ample} implies $A'_0 = \bigoplus_{d \geq 0} H^0(X'_0, (\mathcal{L}'_0)^{\otimes d})$ is a finitely generated graded $R'_0$-algebra and implies there exists a $d'_0$ such that $H^p(X'_0, (\mathcal{L}'_0)^{\otimes d}) = 0$, $p > 0$ for $d \geq d'_0$. By the arguments given above applied to $X'_0 \to \Spec(R'_0)$ and $\mathcal{L}'_0$ we see that $(A'_0)_d$ is a finite projective $R'_0$-module and that $$ A'_d = \mathcal{A}'_d(U') = H^0(X'_{U'}, (\mathcal{L}')^{\otimes d}|_{X'_{U'}}) = H^0(X'_0, (\mathcal{L}'_0)^{\otimes d}) \otimes_{R'_0} R' = (A'_0)_d \otimes_{R'_0} R' $$ for $d \geq d'_0$. Now a small twist in the argument is that we don't know that we can choose $d'_0$ equal to $d_0$\footnote{Actually, one can reduce to this case by doing more limit arguments.}. To get around this we use the following sequence of arguments to finish the proof: \begin{enumerate} \item[(a)] The algebra $B = R'_0 \oplus \bigoplus_{d \geq \max(d_0, d'_0)} (A'_0)_d$ is an $R'_0$-algebra of finite type: apply the Artin-Tate lemma to $B \subset A'_0$, see Algebra, Lemma \ref{algebra-lemma-Artin-Tate}. \item[(b)] As $R'_0$ is Noetherian we see that $B$ is an $R'_0$-algebra of finite presentation. \item[(c)] By right exactness of tensor product we see that $B \otimes_{R'_0} R'$ is an $R'$-algebra of finite presentation. \item[(d)] By the displayed equalities this exactly says that $C = R' \oplus \bigoplus_{d \geq \max(d_0, d'_0)} A'_d$ is an $R'$-algebra of finite presentation. \item[(e)] The quotient $A'/C$ is the direct sum of the finite projective $R'$-modules $A'_d$, $d_0 \leq d \leq \max(d_0, d'_0)$, hence finitely presented as $R'$-module. \item[(f)] The quotient $A'/C$ is finitely presented as a $C$-module by Algebra, Lemma \ref{algebra-lemma-finitely-presented-over-subring}. \item[(g)] Thus $A'$ is finitely presented as a $C$-module by Algebra, Lemma \ref{algebra-lemma-extension}. \item[(h)] By Algebra, Lemma \ref{algebra-lemma-finite-finite-type} this implies $A'$ is finitely presented as a $C$-algebra. \item[(i)] Finally, by Algebra, Lemma \ref{algebra-lemma-compose-finite-type} applied to $R' \to C \to A'$ this implies $A'$ is finitely presented as an $R'$-algebra. \end{enumerate} This finishes the proof.