An associated sheaf restricts to an associated sheaf on an affine open

The family $\mathcal{B}=\{D(f):f\in A,D(f)\subseteq W\}$ is a basis for the topology of $W$: indeed, let $U\subseteq W$ be open and $w\in U$; then $U$ is open in $X$, so by [F1] there is $f\in A$ with $w\in D(f)\subseteq U$, and then $D(f)\subseteq U\subseteq W$, so $D(f)\in \mathcal{B}$ and $w\in D(f)\subseteq U$ as required.

For $D(f)\in \mathcal{B}$ the two sheaves have canonically identified sections: by [F3], $D_{\mathrm{Spec}C}(\phi (f))=D(f)\cap W=D(f)$, and by [F2] applied to this common open set the rings $\mathrm{\Gamma }(D(f),{\mathcal{O}}_X)=A_f$ and $\mathrm{\Gamma }(D_{\mathrm{Spec}C}(\phi (f)),{\mathcal{O}}_W)=C_{\phi (f)}$ are the same ring under restriction, so the inverse ${\alpha }_f:C_{\phi (f)}\to A_f$ of that restriction is an isomorphism of $A$-algebras, because [F4] identifies the two rings with the sections of the unique structure sheaf on this open set and both receive $A$ by the canonical map, which on the $C$ side is $\phi$ followed by localisation; hence ${\alpha }_f$ induces an isomorphism of $A$-modules ${\beta }_f:\mathrm{\Gamma }(D_{\mathrm{Spec}C}(\phi (f)),\widetilde{(C{\otimes }_{A}M)})=(C{\otimes }_{A}M{)}_{\phi (f)}=C_{\phi (f)}{\otimes }_{A}M\to A_f{\otimes }_{A}M=M_f=\mathrm{\Gamma }(D(f),\tilde{M})$, using [F5] on both sides.

General basis-gluing step: let $\mathcal{F},\mathcal{G}$ be sheaves of modules on a space $Z$ and let ${\mathcal{B}}_Z$ be a basis of $Z$, and suppose given for each $B\in {\mathcal{B}}_Z$ an isomorphism of modules ${\beta }_B:\mathcal{F}(B)\to \mathcal{G}(B)$ with ${\beta }_{B^{\mathrm{\prime }}}(s{\mathrm{\mid }}_{B^{\mathrm{\prime }}})={\beta }_B(s){\mathrm{\mid }}_{B^{\mathrm{\prime }}}$ for all $s\in \mathcal{F}(B)$, $B^{\mathrm{\prime }}\in {\mathcal{B}}_Z$, $B^{\mathrm{\prime }}\subseteq B$; then there is a unique isomorphism of sheaves $\beta :\mathcal{F}\to \mathcal{G}$ restricting to ${\beta }_B$ on each $B\in {\mathcal{B}}_Z$: indeed, for open $U\subseteq Z$ and $s\in \mathcal{F}(U)$ the sections ${\beta }_B(s{\mathrm{\mid }}_B)\in \mathcal{G}(B)$, $B\in {\mathcal{B}}_Z$, $B\subseteq U$, are compatible, since for $B,B^{\mathrm{\prime }}\in {\mathcal{B}}_Z$ contained in $U$ and any basis element $B^{\mathrm{\prime }\mathrm{\prime }}\subseteq B\cap B^{\mathrm{\prime }}$ one has ${\beta }_B(s{\mathrm{\mid }}_B){\mathrm{\mid }}_{B^{\mathrm{\prime }\mathrm{\prime }}}={\beta }_{B^{\mathrm{\prime }\mathrm{\prime }}}(s{\mathrm{\mid }}_{B^{\mathrm{\prime }\mathrm{\prime }}})={\beta }_{B^{\mathrm{\prime }}}(s{\mathrm{\mid }}_{B^{\mathrm{\prime }}}){\mathrm{\mid }}_{B^{\mathrm{\prime }\mathrm{\prime }}}$ by the hypothesis and these $B^{\mathrm{\prime }\mathrm{\prime }}$ cover $B\cap B^{\mathrm{\prime }}$, so [F6] gives equality on $B\cap B^{\mathrm{\prime }}$, and by [F6] they glue to a unique ${\beta }_U(s)\in \mathcal{G}(U)$; the maps ${\beta }_U$ are compatible with restrictions, because for $U^{\mathrm{\prime }}\subseteq U$ the sections ${\beta }_U(s){\mathrm{\mid }}_{U^{\mathrm{\prime }}}$ and ${\beta }_{U^{\mathrm{\prime }}}(s{\mathrm{\mid }}_{U^{\mathrm{\prime }}})$ agree after restriction to every basis element $B\subseteq U^{\mathrm{\prime }}$, hence on $U^{\mathrm{\prime }}$ by [F6]; applying the same construction to the inverses ${\beta }_B^{-1}$ produces maps ${\gamma }_U:\mathcal{G}(U)\to \mathcal{F}(U)$, and both ${\gamma }_U\circ {\beta }_U$ and ${\beta }_U\circ {\gamma }_U$ are the identities because they agree on every basis element of $U$ and hence on $U$ by [F6], while uniqueness follows since a morphism is determined by its components on a basis cover by [F6].

The identifications ${\beta }_f$ of step 1.2 are compatible with restrictions along $D(g)\subseteq D(f)$ with $D(f),D(g)\in \mathcal{B}$: by [F4] and [F5] the restriction of $\tilde{M}$ is $M_f\to M_g$ and that of $\widetilde{(C{\otimes }_{A}M)}$ is $(C{\otimes }_{A}M{)}_{\phi (f)}\to (C{\otimes }_{A}M{)}_{\phi (g)}$, both induced by the restriction maps of structure sheaves, and under the ring identifications ${\alpha }_f,{\alpha }_g$ of step 1.2 these two maps correspond because both structure-sheaf restrictions $\mathrm{\Gamma }(D(f),{\mathcal{O}}_X)=A_f\to A_g=\mathrm{\Gamma }(D(g),{\mathcal{O}}_X)$ and $C_{\phi (f)}\to C_{\phi (g)}$ are the restriction of the same sheaf ${\mathcal{O}}_X={\mathcal{O}}_W$ along $D(g)\subseteq D(f)$, so tensoring over $A$ with $M$ gives compatibility; the assignment is natural in $M$ because a map $u:M\to N$ induces components $u_f$ and $(C\otimes u{)}_{\phi (f)}$ compatible with the $\alpha$'s by [F5].

Applying step 1.3 with $Z=W$, the basis $\mathcal{B}$ of step 1.1 and the isomorphisms ${\beta }_f$ of steps 1.2 and 2.1 yields a canonical isomorphism of ${\mathcal{O}}_W$-modules $\widetilde{(C{\otimes }_{A}M)}\to (\tilde{M}){\mathrm{\mid }}_W$ restricting to ${\beta }_f$ on $D(f)\in \mathcal{B}$, whose inverse is the isomorphism $(\tilde{M}){\mathrm{\mid }}_W\cong \widetilde{(C{\otimes }_{A}M)}$ of the Statement; it is natural in $M$ by the naturality recorded in steps 1.2 and 2.1, and the Axiom of Choice ([F7]) enters only through [F5], which supplies the two associated sheaves, while the comparison chooses nothing because the basis $\mathcal{B}$ is determined by $A$ and $W$ and the identifications ${\beta }_f$ are canonical.