Universal property of relative differential sheaves

Sheafification adjunction. Restriction along $P\to aP={\mathrm{\Omega }}_{X\mathrm{/}S}$ gives a bijection ${\mathrm{Hom}}_{{\mathcal{O}}_X}({\mathrm{\Omega }}_{X\mathrm{/}S},\mathcal{F})\cong {\mathrm{Hom}}_{{\mathcal{O}}_X\text{-presheaf}}(P,\mathcal{F})$: [F2] gives the factorization on underlying presheaves, and the factor is ${\mathcal{O}}_X$-linear because sections of $aP$ are locally represented by sections of $P$, with scalar multiplication defined on those representatives by [F1]. Linearity therefore holds locally and hence globally. A morphism of presheaves $P\to \mathcal{F}$ is exactly a compatible family of ${\mathcal{O}}_X(W)$-linear maps ${\phi }_W\text{\hspace{0.17em}⁣}:P(W)\to \mathcal{F}(W)$.

Algebraic universal property on each open. Since ${\mathcal{O}}_X(W)$ is an $(f^{-1}{\mathcal{O}}_S)(W)$-algebra, [F3] turns ${\phi }_W$ into the derivation $D_W:={\phi }_W\circ {\mathrm{d}}_W\text{\hspace{0.17em}⁣}:{\mathcal{O}}_X(W)\to \mathcal{F}(W)$, an $(f^{-1}{\mathcal{O}}_S)(W)$-derivation by [F5]; conversely every such derivation arises from exactly one ${\phi }_W$. Compatibility of the family $({\phi }_W)$ under restriction is equivalent to compatibility of the family $(D_W)$, because the restriction maps of ${\mathrm{\Omega }}_{X\mathrm{/}S}$ are defined so that ${\mathrm{d}}_{W^{\mathrm{\prime }}}\circ \rho =\rho \circ {\mathrm{d}}_W$ for $W^{\mathrm{\prime }}\subseteq W$.

Compatible families of derivations are $S$-derivations. The families $(D_W)$ in step 1.2 are in canonical bijection with morphisms of sheaves of abelian groups $D\text{\hspace{0.17em}⁣}:{\mathcal{O}}_X\to \mathcal{F}$ that are additive and satisfy Leibniz on every open and kill the image of each $(f^{-1}{\mathcal{O}}_S)(W)$; by gluing, the last condition is exactly $D\circ f^{\mathrm{♯}}=0$, so these are precisely the $S$-derivations of [F1]. The two passes are inverse because a derivation determines its components $D_W$, and ${\phi }_W$ is recovered from $D_W$ by the universal property of $P(W)$.

Conclusion. Composing the bijections of steps 1.1, 1.2 and 2.1 gives the displayed bijection $\alpha \mapsto \alpha \circ {\mathrm{d}}_{X\mathrm{/}S}$, since $\alpha$ corresponds to the composite of its components with ${\mathrm{d}}_W$ and ${\mathrm{d}}_{X\mathrm{/}S}$ is assembled from the ${\mathrm{d}}_W$ by [F1]. Each step is natural in $\mathcal{F}$: a morphism $\mathcal{F}\to \mathcal{G}$ of ${\mathcal{O}}_X$-modules composes with ${\phi }_W$ and with $D_W$, so the bijection is compatible with postcomposition, and by the Yoneda lemma ${\mathrm{\Omega }}_{X\mathrm{/}S}$ is determined up to unique compatible isomorphism.