First, assume that $X$ is locally Noetherian. By Lemma \ref{lemma-Garel-upstairs} we have a canonical map $$ c^p_{Y/X} : \Omega_{Y/S}^p \longrightarrow f^*\Omega_{X/S}^p \otimes_{\mathcal{O}_Y} \det(\NL_{Y/X}) $$ By Discriminants, Proposition \ref{discriminant-proposition-tate-map} we have a canonical isomorphism $$ c_{Y/X} : \det(\NL_{Y/X}) \to \omega_{Y/X} $$ mapping $\delta(\NL_{Y/X})$ to $\tau_{Y/X}$. Combined these maps give $$ c^p_{Y/X} \otimes c_{Y/X} : \Omega_{Y/S}^p \longrightarrow f^*\Omega_{X/S}^p \otimes_{\mathcal{O}_Y} \omega_{Y/X} $$ By Discriminants, Section \ref{discriminant-section-finite-morphisms} this is the same thing as a map $$ \Theta_{Y/X}^p : f_*\Omega_{Y/S}^p \longrightarrow \Omega_{X/S}^p $$ Recall that the relationship between $c^p_{Y/X} \otimes c_{Y/X}$ and $\Theta_{Y/X}^p$ uses the evaluation map $f_*\omega_{Y/X} \to \mathcal{O}_X$ which sends $\tau_{Y/X}$ to $\text{Trace}_f(1)$, see Discriminants, Section \ref{discriminant-section-finite-morphisms}. Hence property (1) holds. Property (2) holds for base changes by $X' \to X$ with $X'$ locally Noetherian because both $c^p_{Y/X}$ and $c_{Y/X}$ are compatible with such base changes. For $f : Y \to X$ finite syntomic and $X$ locally Noetherian, we will continue to denote $\Theta^p_{Y/X}$ the solution we've just found. \medskip\noindent Uniqueness. Suppose that we have a finite syntomic morphism $f: Y \to X$ such that $X$ is smooth over $\Spec(\mathbf{Z})$ and $f$ is \'etale over a dense open of $X$. We claim that in this case $\Theta^p_{Y/X}$ is uniquely determined by property (1). Namely, consider the maps $$ \Omega^p_{X/\mathbf{Z}} \otimes_{\mathcal{O}_X} f_*\mathcal{O}_Y \to f_*\Omega^p_{Y/\mathbf{Z}} \to \Omega^p_{X/\mathbf{Z}} $$ The sheaf $\Omega^p_{X/\mathbf{Z}}$ is torsion free (by the assumed smoothness), hence it suffices to check that the restriction of $\Theta^p_{Y/X}$ is uniquely determined over the dense open over which $f$ is \'etale, i.e., we may assume $f$ is \'etale. However, if $f$ is \'etale, then $f^*\Omega_{X/\mathbf{Z}} = \Omega_{Y/\mathbf{Z}}$ hence the first arrow in the displayed equation is an isomorphism. Since we've pinned down the composition, this guarantees uniqueness. \medskip\noindent Let $f : Y \to X$ be a finite syntomic morphism of locally Noetherian schemes. Let $x \in X$. By Discriminants, Lemma \ref{discriminant-lemma-locally-comes-from-universal-finite} we can find $d \geq 1$ and a commutative diagram $$ \xymatrix{ Y \ar[d] & V \ar[d] \ar[l] \ar[r] & V_d \ar[d] \\ X & U \ar[l] \ar[r] & U_d } $$ such that $x \in U \subset X$ is open, $V = f^{-1}(U)$ and $V = U \times_{U_d} V_d$. Thus $\Theta^p_{Y/X}|_V$ is the pullback of the map $\Theta^p_{V_d/U_d}$. However, by the discussion on uniqueness above and Discriminants, Lemmas \ref{discriminant-lemma-universal-finite-syntomic-smooth} and \ref{discriminant-lemma-universal-finite-syntomic-etale} the map $\Theta^p_{V_d/U_d}$ is uniquely determined by the requirement (1). Hence uniqueness holds. \medskip\noindent At this point we know that we have existence and uniqueness for all finite syntomic morphisms $Y \to X$ with $X$ locally Noetherian. We could now give an argument similar to the proof of Lemma \ref{lemma-Garel-upstairs} to extend to general $X$. However, instead it possible to directly use absolute Noetherian approximation to finish the proof. Namely, to construct $\Theta^p_{Y/X}$ it suffices to do so Zariski locally on $X$ (provided we also show the uniqueness). Hence we may assume $X$ is affine (small detail omitted). Then we can write $X = \lim_{i \in I} X_i$ as the limit over a directed set $I$ of Noetherian affine schemes. By Algebra, Lemma \ref{algebra-lemma-colimit-category-fp-algebras} we can find $0 \in I$ and a finitely presented morphism of affines $f_0 : Y_0 \to X_0$ whose base change to $X$ is $Y \to X$. After increasing $0$ we may assume $Y_0 \to X_0$ is finite and syntomic, see Algebra, Lemma \ref{algebra-lemma-colimit-lci} and \ref{algebra-lemma-colimit-finite}. For $i \geq 0$ also the base change $f_i : Y_i = Y_0 \times_{X_0} X_i \to X_i$ is finite syntomic. Then $$ \Gamma(X, f_*\Omega^p_{Y/\mathbf{Z}}) = \Gamma(Y, \Omega^p_{Y/\mathbf{Z}}) = \colim_{i \geq 0} \Gamma(Y_i, \Omega^p_{Y_i/\mathbf{Z}}) = \colim_{i \geq 0} \Gamma(X_i, f_{i, *}\Omega^p_{Y_i/\mathbf{Z}}) $$ Hence we can (and are forced to) define $\Theta^p_{Y/X}$ as the colimit of the maps $\Theta^p_{Y_i/X_i}$. This map is compatible with any cartesian diagram $$ \xymatrix{ Y' \ar[r] \ar[d] & Y \ar[d] \\ X' \ar[r] & X } $$ with $X'$ affine as we know this for the case of Noetherian affine schemes by the arguments given above (small detail omitted; hint: if we also write $X' = \lim_{j \in J} X'_j$ then for every $i \in I$ there is a $j \in J$ and a morphism $X'_j \to X_i$ compatible with the morphism $X' \to X$). This finishes the proof.