Symmetric algebras are quasi-coherent and commute with pullback

Universal property of the affine model: for a commutative ring $A$, an $A$-module $M$ and a commutative $A$-algebra $C$, the map $g\mapsto g\circ \eta$ induced by the degree-one inclusion $\eta :M\to {\mathrm{Sym}}_A(M)$ is a bijection $A\text{-}\mathrm{A}\mathrm{l}\mathrm{g}({\mathrm{Sym}}_A(M),C)\to {\mathrm{Hom}}_A(M,C)$: apply the universal property of [F1] on $X=\mathrm{Spec}A$ to $\mathcal{F}=\tilde{M}$ and the sheaf of commutative ${\mathcal{O}}_X$-algebras $\tilde{C}$, and translate along the affine equivalence [F2], under which ${\mathcal{O}}_X$-linear maps $\tilde{M}\to \tilde{C}$ are the $A$-linear maps $M\to C$, while $\mathrm{Sym}(\tilde{M})\cong ({\mathrm{Sym}}_A(M){)}^{\sim }$ by the affine model of [F1] and the tensor identifications [F3] match the multiplications, so ${\mathcal{O}}_X$-algebra maps $\mathrm{Sym}(\tilde{M})\to \tilde{C}$ are the $A$-algebra maps ${\mathrm{Sym}}_A(M)\to C$; the bijection is natural in $C$ because the equivalence and the universal property are.

Base change of the affine model: for a ring map $A\to B$ and an $A$-module $M$ there is a canonical isomorphism of graded $B$-algebras $\beta :B{\otimes }_A{\mathrm{Sym}}_A(M)\to {\mathrm{Sym}}_B(B{\otimes }_{A}M)$ characterized by $\beta (1\otimes {\eta }_A(m))={\eta }_B(1\otimes m)$: both sides represent the same functor of commutative $B$-algebras, since [F6] identifies $B$-algebra maps $B{\otimes }_A{\mathrm{Sym}}_A(M)\to D$ with $A$-algebra maps ${\mathrm{Sym}}_A(M)\to D$, step 1.1 with $A$-linear maps $M\to D$, [F6] with $B$-linear maps $B{\otimes }_{A}M\to D$, and step 1.1 over $B$ with $B$-algebra maps ${\mathrm{Sym}}_B(B{\otimes }_{A}M)\to D$; the identity of ${\mathrm{Sym}}_B(B{\otimes }_{A}M)$ corresponds to a map $\beta$ and the identity of $B{\otimes }_A{\mathrm{Sym}}_A(M)$ to a map $\gamma$, and naturality in $D$ gives $\beta \circ \gamma =\mathrm{i}\mathrm{d}$ and $\gamma \circ \beta =\mathrm{i}\mathrm{d}$. Under this correspondence $\beta$ is the unique $B$-algebra map with $\beta (1\otimes {\eta }_A(m))={\eta }_B(1\otimes m)$, so it preserves the degree-one generators and hence the grading, and for $A\to B\to C$ both composites $C{\otimes }_A{\mathrm{Sym}}_A(M)\to {\mathrm{Sym}}_C(C{\otimes }_{A}M)$ in the claim are $C$-algebra maps sending $1\otimes {\eta }_A(m)$ to ${\eta }_C(1\otimes m)$; since the source is generated as a $C$-algebra by these elements, the composites agree, the identifications $C{\otimes }_B(B{\otimes }_{A}N)\cong C{\otimes }_{A}N$ being those of [F6].

Restriction to open subschemes: for an open subscheme $W\subseteq X$ there is a canonical isomorphism of graded ${\mathcal{O}}_W$-algebras $\mathrm{Sym}(\mathcal{F}){\mathrm{\mid }}_W\cong \mathrm{Sym}(\mathcal{F}{\mathrm{\mid }}_W)$, compatible with inclusions $W^{\mathrm{\prime }}\subseteq W$: the affine opens $U=\mathrm{Spec}A\subseteq W$ with $\mathcal{F}{\mathrm{\mid }}_U\cong \tilde{M}$ cover $W$ because $\mathcal{F}$ is quasi-coherent, and on each of them $\mathrm{Sym}(\mathcal{F}){\mathrm{\mid }}_U\cong ({\mathrm{Sym}}_A(M){)}^{\sim }$ by the affine model, while applying the same model to the quasi-coherent sheaf $\mathcal{F}{\mathrm{\mid }}_W$ on $W$ gives $\mathrm{Sym}(\mathcal{F}{\mathrm{\mid }}_W){\mathrm{\mid }}_U\cong ({\mathrm{Sym}}_A(M){)}^{\sim }$ with the same module $M=\mathrm{\Gamma }(U,\mathcal{F})$; for affine $U^{\mathrm{\prime }}\subseteq U$ both identifications restrict by the canonical base-change isomorphism of step 2.1 for the ring map $\mathrm{\Gamma }(U,\mathcal{O})\to \mathrm{\Gamma }(U^{\mathrm{\prime }},\mathcal{O})$, so the chartwise isomorphisms agree on overlaps and glue by [F5] to a graded ${\mathcal{O}}_W$-algebra isomorphism, and passing to the subcover inside $W^{\mathrm{\prime }}\subseteq W$ shows compatibility with inclusions.

Pullback along an arbitrary morphism: for $f:Y\to X$ there is a canonical isomorphism of graded ${\mathcal{O}}_Y$-algebras $f^{\ast }\mathrm{Sym}(\mathcal{F})\cong \mathrm{Sym}(f^{\ast }\mathcal{F})$: cover $Y$ by affine opens $V=\mathrm{Spec}B$ with $f(V)\subseteq U=\mathrm{Spec}A$ for an affine $U\subseteq X$ on which $\mathcal{F}{\mathrm{\mid }}_U\cong \tilde{M}$, which is possible because affine opens form a basis and $\mathcal{F}$ is quasi-coherent; on such $V$ the affine form of the pullback [F4] applied to $\mathcal{F}$ and to the quasi-coherent $\mathrm{Sym}(\mathcal{F})$ gives $f^{\ast }\mathcal{F}{\mathrm{\mid }}_V\cong \widetilde{(B{\otimes }_{A}M)}$ and $f^{\ast }\mathrm{Sym}(\mathcal{F}){\mathrm{\mid }}_V\cong \widetilde{(B{\otimes }_A{\mathrm{Sym}}_A(M))}$, while the affine model [F1] applied to the quasi-coherent $f^{\ast }\mathcal{F}$ on $V$ gives $\mathrm{Sym}(f^{\ast }\mathcal{F}){\mathrm{\mid }}_V\cong ({\mathrm{Sym}}_B\mathrm{\Gamma }(V,f^{\ast }\mathcal{F}){)}^{\sim }=({\mathrm{Sym}}_B(B{\otimes }_{A}M){)}^{\sim }$ using $\mathrm{\Gamma }(V,\widetilde{(B{\otimes }_{A}M)})=B{\otimes }_{A}M$ [F2]; the base-change isomorphism $\beta$ of step 2.1 is a graded ${\mathcal{O}}_V$-algebra isomorphism between these two models, and on an affine $V^{\mathrm{\prime }}\subseteq V$ the two restrictions agree because $\beta$ is compatible with the ring map $B\to \mathrm{\Gamma }(V^{\mathrm{\prime }},\mathcal{O})$ by step 2.1, on overlaps from different target charts they also agree: both send each local pulled-back degree-one section to its degree-one image, and such sections generate the source algebra. Thus the local isomorphisms glue over the cover of $Y$ by [F5] to a canonical isomorphism which is graded and multiplicative because the local ones are.

The remaining claims: $\mathrm{Sym}(\mathcal{F})$ is a quasi-coherent graded commutative ${\mathcal{O}}_X$-algebra with ${\mathrm{Sym}}^0(\mathcal{F})={\mathcal{O}}_X$ and ${\mathrm{Sym}}^1(\mathcal{F})=\mathcal{F}$, and for $\mathcal{F}={\mathcal{O}}_X^r$ it is ${\mathcal{O}}_X[T_1,\dots ,T_r]$ with $\mathrm{deg}{T}_i=1$, by the affine models of [F1]; the base-change isomorphism of step 2.1 for $M=A^r$ gives the same polynomial algebra after extension of scalars, so the two descriptions are consistent.

Choice and degenerate cases: all identifications in steps 1.1 to 3.3 are canonical, being induced by the canonical base-change isomorphisms and universal properties, so no chart, module or isomorphism is selected and the only Axiom of Choice is the inherited one recorded in [F7], including the compactness use in the affine equivalence [F2] invoked in steps 1.1 and 3.2; for $\mathcal{F}=0$ one has $\mathrm{Sym}(\mathcal{F})={\mathcal{O}}_X$ concentrated in degree zero and $\beta$ is the canonical identification $B{\otimes }_{A}A\cong B$, and for $X=\mathrm{\varnothing }$ both sides of every statement are the zero sheaf.