We first establish the existence of a couple of ``global'' sheaves and global maps of sheaves, and further down we describe the constructions over some affine opens. \medskip\noindent Recall that $\Delta = \Delta_{X/S} : X \to X \times_S X$ is an immersion, see Schemes, Lemma \ref{schemes-lemma-diagonal-immersion}. Let $\mathcal{J}$ be the ideal sheaf of the immersion which lives over some open subscheme $W$ of $X \times_S X$ such that $\Delta(X) \subset W$ is closed. Let us take the one that was found in the proof of Schemes, Lemma \ref{schemes-lemma-diagonal-immersion}. Note that the sheaf of rings $\mathcal{O}_W/\mathcal{J}^2$ is supported on $\Delta(X)$. Moreover it sits in a short exact sequence of sheaves $$ 0 \to \mathcal{J}/\mathcal{J}^2 \to \mathcal{O}_W/\mathcal{J}^2 \to \Delta_*\mathcal{O}_X \to 0. $$ Using $\Delta^{-1}$ we can think of this as a surjection of sheaves of $f^{-1}\mathcal{O}_S$-algebras with kernel the conormal sheaf of $\Delta$ (see Definition \ref{definition-conormal-sheaf} and Lemma \ref{lemma-affine-conormal}). $$ 0 \to \mathcal{C}_{X/X \times_S X} \to \Delta^{-1}(\mathcal{O}_W/\mathcal{J}^2) \to \mathcal{O}_X \to 0 $$ This places us in the situation of Modules, Lemma \ref{modules-lemma-double-structure-gives-derivation}. The projection morphisms $p_i : X \times_S X \to X$, $i = 1, 2$ induce maps of sheaves of rings $(p_i)^\sharp : (p_i)^{-1}\mathcal{O}_X \to \mathcal{O}_{X \times_S X}$. We may restrict to $W$ and quotient by $\mathcal{J}^2$ to get $(p_i)^{-1}\mathcal{O}_X \to \mathcal{O}_W/\mathcal{J}^2$. Since $\Delta^{-1}p_i^{-1}\mathcal{O}_X = \mathcal{O}_X$ we get maps $$ s_i : \mathcal{O}_X \to \Delta^{-1}(\mathcal{O}_W/\mathcal{J}^2). $$ Both $s_1$ and $s_2$ are sections to the map $\Delta^{-1}(\mathcal{O}_W/\mathcal{J}^2) \to \mathcal{O}_X$, as in Modules, Lemma \ref{modules-lemma-double-structure-gives-derivation}. Thus we get an $S$-derivation $\text{d} = s_2 - s_1 : \mathcal{O}_X \to \mathcal{C}_{X/X \times_S X}$. By the universal property of the module of differentials we find a unique $\mathcal{O}_X$-linear map $$ \Omega_{X/S} \longrightarrow \mathcal{C}_{X/X \times_S X},\quad f\text{d}g \longmapsto fs_2(g) - fs_1(g) $$ To see the map is an isomorphism, let us work this out over suitable affine opens. We can cover $X$ by affine opens $\Spec(A) = U \subset X$ whose image is contained in an affine open $\Spec(R) = V \subset S$. According to the proof of Schemes, Lemma \ref{schemes-lemma-diagonal-immersion} $U \times_V U \subset X \times_S X$ is an affine open contained in the open $W$ mentioned above. Also $U \times_V U = \Spec(A \otimes_R A)$. The sheaf $\mathcal{J}$ corresponds to the ideal $J = \Ker(A \otimes_R A \to A)$. The short exact sequence to the short exact sequence of $A \otimes_R A$-modules $$ 0 \to J/J^2 \to (A \otimes_R A)/J^2 \to A \to 0 $$ The sections $s_i$ correspond to the ring maps $$ A \longrightarrow (A \otimes_R A)/J^2,\quad s_1 : a \mapsto a \otimes 1,\quad s_2 : a \mapsto 1 \otimes a. $$ By Lemma \ref{lemma-affine-conormal} we have $\Gamma(U, \mathcal{C}_{X/X \times_S X}) = J/J^2$ and by Lemma \ref{lemma-differentials-affine} we have $\Gamma(U, \Omega_{X/S}) = \Omega_{A/R}$. The map above is the map $a \text{d}b \mapsto a \otimes b - ab \otimes 1$ which is shown to be an isomorphism in Algebra, Lemma \ref{algebra-lemma-differentials-diagonal}.