By Discriminants, Lemma \ref{discriminant-lemma-locally-comes-from-universal-finite} for every $x \in X$ we can find $d \geq 1$ and a commutative diagram $$ \xymatrix{ Y \ar[d] & V \ar[d] \ar[l] \ar[r] & V_d \ar[d] \ar[r] & Y_d = \Spec(B_d) \ar[d] \\ X & U \ar[l] \ar[r] & U_d \ar[r] & X_d = \Spec(A_d) } $$ such that $x \in U \subset X$ is affine open, $V = f^{-1}(U)$ and $V = U \times_{U_d} V_d$. Write $U = \Spec(A)$ and $V = \Spec(B)$ and observe that $B = A \otimes_{A_d} B_d$ and recall that $B_d = A_d e_1 \oplus \ldots \oplus A_d e_d$. Suppose we have $a_1, \ldots, a_r \in A$ and $b_1, \ldots, b_s \in B$. We may write $b_j = \sum a_{j, l} e_d$ with $a_{j, l} \in A$. Set $N = r + sd$ and consider the factorizations $$ \xymatrix{ V \ar[r] \ar[d] & V' = \mathbf{A}^N \times V_d \ar[r] \ar[d] & V_d \ar[d] \\ U \ar[r]& U' = \mathbf{A}^N \times U_d \ar[r] & U_d } $$ Here the horizontal lower right arrow is given by the morphism $U \to U_d$ (from the earlier diagram) and the morphism $U \to \mathbf{A}^N$ given by $a_1, \ldots, a_r, a_{1, 1}, \ldots, a_{s, d}$. Then we see that the functions $a_1, \ldots, a_r$ are in the image of $\Gamma(U', \mathcal{O}_{U'}) \to \Gamma(U, \mathcal{O}_U)$ and the functions $b_1, \ldots, b_s$ are in the image of $\Gamma(V', \mathcal{O}_{V'}) \to \Gamma(V, \mathcal{O}_V)$. In this way we see that for any finite collection of elements\footnote{After all these elements will be finite sums of elements of the form $a_0 \text{d}a_1 \wedge \ldots \wedge \text{d}a_i$ with $a_0, \ldots, a_i \in A$ or finite sums of elements of the form $b_0 \text{d}b_1 \wedge \ldots \wedge \text{d}b_j$ with $b_0, \ldots, b_j \in B$.} of the groups $$ \Gamma(V, \Omega^i_{Y/\mathbf{Z}}),\quad i = 0, 1, 2, \ldots \quad\text{and}\quad \Gamma(U, \Omega^j_{X/\mathbf{Z}}),\quad j = 0, 1, 2, \ldots $$ we can find a factorizations $V \to V' \to V_d$ and $U \to U' \to U_d$ with $V' = \mathbf{A}^N \times V_d$ and $U' = \mathbf{A}^N \times U_d$ as above such that these sections are the pullbacks of sections from $$ \Gamma(V', \Omega^i_{V'/\mathbf{Z}}),\quad i = 0, 1, 2, \ldots \quad\text{and}\quad \Gamma(U', \Omega^j_{U'/\mathbf{Z}}),\quad j = 0, 1, 2, \ldots $$ The upshot of this is that to check $\text{d} \circ \Theta_{Y/X} = \Theta_{Y/X} \circ \text{d}$ it suffices to check this is true for $\Theta_{V'/U'}$. Similarly, for property (2) of the lemma. \medskip\noindent By Discriminants, Lemmas \ref{discriminant-lemma-universal-finite-syntomic-smooth} and \ref{discriminant-lemma-universal-finite-syntomic-etale} the scheme $U_d$ is smooth and the morphism $V_d \to U_d$ is \'etale over a dense open of $U_d$. Hence the same is true for the morphism $V' \to U'$. Since $\Omega_{U'/\mathbf{Z}}$ is locally free and hence $\Omega^p_{U'/\mathbf{Z}}$ is torsion free, it suffices to check the desired relations after restricting to the open over which $V'$ is finite \'etale. Then we may check the relations after a surjective \'etale base change. Hence we may split the finite \'etale cover and assume we are looking at a morphism of the form $$ \coprod\nolimits_{i = 1, \ldots, d} W \longrightarrow W $$ with $W$ smooth over $\mathbf{Z}$. In this case any local properties of our construction are trivial to check (provided they are true). This finishes the proof of (1) and (2). \medskip\noindent Finally, we observe that (3) follows from (2) because $\Omega_{Y/S}$ is the quotient of $\Omega_{Y/\mathbf{Z}}$ by the submodule generated by pullbacks of local sections of $\Omega_{S/\mathbf{Z}}$.