Affine charts recover the algebraic module of differentials

Restriction of derivations. Let $D\text{\hspace{0.17em}⁣}:{\mathcal{O}}_X\to \mathcal{F}$ be an $S$-derivation. Its global component $D_X\text{\hspace{0.17em}⁣}:B\to \mathcal{F}(X)$ is additive and satisfies Leibniz, and it kills the image of $A$, because $A$ maps into ${\mathcal{O}}_X(X)=B$ through $(f^{-1}{\mathcal{O}}_S)(X)$ and $D$ kills that image by [F7]. So $D\mapsto D_X$ is a map ${\mathrm{Der}}_S({\mathcal{O}}_X,\mathcal{F})\to {\mathrm{Der}}_A(B,\mathcal{F}(X))$.

Localizing a derivation of global sections. Conversely let $D_0\text{\hspace{0.17em}⁣}:B\to \mathcal{F}(X)$ be an $A$-derivation. For every $g$ the composite $B\stackrel{D_0}{\to }\mathcal{F}(X)\to \mathcal{F}(D(g))$ is an $A$-derivation, so by [F3] it corresponds to a $B$-linear map ${\mathrm{\Omega }}_{B\mathrm{/}A}\to \mathcal{F}(D(g))$, which by [F4] is the same as a $B_g$-linear map ${\epsilon }_g\text{\hspace{0.17em}⁣}:{\mathrm{\Omega }}_{B_g\mathrm{/}A}\to \mathcal{F}(D(g))$; put $D_g:={\epsilon }_g\circ {\mathrm{d}}_{B_g\mathrm{/}A}\text{\hspace{0.17em}⁣}:B_g\to \mathcal{F}(D(g))$. These maps are compatible with restriction to a smaller basic open, since both restrictions are induced by the same $A$-derivation composite $B\to \mathcal{F}(D(gh))$ and [F4] is compatible with the universal derivations.

Gluing. For an open $W\subseteq X$ and $a\in {\mathcal{O}}_X(W)$, the elements $D_g(a{\mathrm{\mid }}_{D(g)})$, indexed by basic opens $D(g)\subseteq W$, are compatible on intersections $D(gh)$ by step 1.2, so they glue to a unique element $D_W(a)\in \mathcal{F}(W)$. The resulting $D_W$ are additive and satisfy Leibniz because this can be checked on a basic-open cover. They kill $f^{-1}{\mathcal{O}}_S$ locally: a germ in the image of $f^{-1}{\mathcal{O}}_S$ at $x\in D(g)$ comes from a section of ${\mathcal{O}}_S$ on an open neighbourhood of $f(x)$; after shrinking to an affine neighbourhood of $f(x)$ and then to a basic open around $x$, that section is a fraction of elements of $A$. The derivation $D_g$ kills $A$, and the Leibniz rule applied to an inverse shows it kills such fractions. Vanishing at every stalk implies the sheaf composite $f^{-1}{\mathcal{O}}_S\to \mathcal{F}$ is zero.

The two constructions are inverse. If $D$ is an $S$-derivation with global component $D_0=D_X$, then for each $g$ the map ${\epsilon }_g$ of step 1.2 is the composite ${\mathrm{\Omega }}_{B\mathrm{/}A}\to {\mathrm{\Omega }}_{B_g\mathrm{/}A}\to \mathcal{F}(D(g))$ induced by $D_0$ and restriction, so $D_g$ agrees with $D$ on $B_g$; by the sheaf property, the derivation produced in step 2.1 equals $D$. Conversely the derivation produced from $D_0$ has global component $D_0$, since its component on $D(g)$ restricts from $D_0$. Hence restriction of global sections is a bijection ${\mathrm{Der}}_S({\mathcal{O}}_X,\mathcal{F})\cong {\mathrm{Der}}_A(B,\mathcal{F}(X)),$ natural in $\mathcal{F}$.

The comparison isomorphism. By [F1] with $M={\mathrm{\Omega }}_{B\mathrm{/}A}$ and [F3], ${\mathrm{Hom}}_{{\mathcal{O}}_X}(\tilde{\mathrm{\Omega }},\mathcal{F})\cong {\mathrm{Hom}}_B({\mathrm{\Omega }}_{B\mathrm{/}A},\mathcal{F}(X))\cong {\mathrm{Der}}_A(B,\mathcal{F}(X))$, and by step 3.1 and [F2] the last group is ${\mathrm{Hom}}_{{\mathcal{O}}_X}({\mathrm{\Omega }}_{X\mathrm{/}S},\mathcal{F})$. All identifications are natural in $\mathcal{F}$, so the Yoneda lemma produces a unique isomorphism $\tilde{\mathrm{\Omega }}\to {\mathrm{\Omega }}_{X\mathrm{/}S}$; tracking the universal elements (the identity of $\tilde{\mathrm{\Omega }}$ corresponds to the derivation $b\mapsto \epsilon (\mathrm{d}b)$ and the identity of ${\mathrm{\Omega }}_{X\mathrm{/}S}$ to ${\mathrm{d}}_{X\mathrm{/}S}$) shows that the isomorphism sends $\epsilon (\mathrm{d}b)$ to ${\mathrm{d}}_{X\mathrm{/}S}(b)$. Naturality in the ring map $A\to B$ follows from the functoriality of [F1] in $M$ and of [F3].

Sections over affine and basic opens. The sheaf attached to ${\mathrm{\Omega }}_{B\mathrm{/}A}$ is computed from its values on the distinguished-open basis: the assignment $D(g)\mapsto {\mathrm{\Omega }}_{B\mathrm{/}A}{\otimes }_B{B}_g={\mathrm{\Omega }}_{B_g\mathrm{/}A}$ with the localization maps as restrictions is a sheaf on the basis (the localization exactness makes fractions glue; see Localisation of a module at a multiplicative subset and [F4]) and extends to the sheaf $\tilde{\mathrm{\Omega }}$ with those values and restrictions, exactly as Sections and restrictions on distinguished opens of an affine scheme records for ${\mathcal{O}}_X$ itself. Hence $\mathrm{\Gamma }(X,\tilde{\mathrm{\Omega }})={\mathrm{\Omega }}_{B\mathrm{/}A}$ and ${\mathrm{\Omega }}_{X\mathrm{/}S}(D(g))\cong {\mathrm{\Omega }}_{B_g\mathrm{/}A}$ under step 4.1, compatible with ${\mathrm{d}}_{X\mathrm{/}S}$ by the characterization of that isomorphism and with the localization maps because those are the restriction maps of $\tilde{\mathrm{\Omega }}$.