The implication (2) $\Rightarrow$ (1) follows from the more general Topologies, Lemma \ref{topologies-lemma-V-covering-universally-submersive} and our definition of h covers. Assume $\coprod X_i \to X$ is universally submersive. We will show that $\{X_i \to X\}$ can be refined by a ph covering; this will suffice by Topologies, Lemma \ref{topologies-lemma-refine-by-ph} and our definition of h coverings. The argument will be the same as the one used in the proof of Lemma \ref{lemma-equivalence-h-v-locally-finite-presentation}. \medskip\noindent Choose a generic flatness stratification $$ X = S \supset S_0 \supset S_1 \supset \ldots \supset S_t = \emptyset $$ as in Lemma \ref{lemma-generic-flatness-stratification} for the finitely presented morphism $$ \coprod\nolimits_{i = 1, \ldots, n} f_i : \coprod\nolimits_{i = 1, \ldots, n} X_i \longrightarrow X $$ We are going to use all the properties of the stratification without further mention. By construction the base change of each $f_i$ to $U_k = S_k \setminus S_{k + 1}$ is flat. Denote $Y_k$ the scheme theoretic closure of $U_k$ in $S_k$. Since $U_k \to S_k$ is a quasi-compact open immersion (all schemes in this paragraph are Noetherian), we see that $U_k \subset Y_k$ is a quasi-compact dense (and scheme theoretically dense) open immersion, see Morphisms, Lemma \ref{morphisms-lemma-quasi-compact-scheme-theoretic-image}. The morphism $\coprod_{k = 0, \ldots, t - 1} Y_k \to X$ is finite surjective, hence $\{Y_k \to X\}$ is a ph covering. By the transitivity property of ph coverings (Topologies, Lemma \ref{topologies-lemma-ph}) it suffices to show that the pullback of the covering $\{X_i \to X\}$ to each $Y_k$ can be refined by a ph covering. This reduces us to the case described in the next paragraph. \medskip\noindent Assume $\coprod X_i \to X$ is universally submersive and there is a dense open $U \subset X$ such that $X_i \times_X U \to U$ is flat for all $i$. By Theorem \ref{theorem-flatten-module} there is a $U$-admissible blowup $X' \to X$ such that the strict transform $f'_i : X'_i \to X'$ of $f_i$ is flat for all $i$. Observe that the projective (hence closed) morphism $X' \to X$ is surjective as $U \subset X$ is dense and as $U$ is identified with an open of $X'$. After replacing $X'$ by a further $U$-admissible blowup if necessary, we may also assume $U \subset X'$ is dense (see Remark \ref{remark-successive-blowups}). Hence for every point $x \in X'$ there is a discrete valuation ring $A$ and a morphism $g : \Spec(A) \to X'$ such that the generic point of $\Spec(A)$ maps into $U$ and the closed point of $\Spec(A)$ maps to $x$, see Limits, Lemma \ref{limits-lemma-reach-point-closure-Noetherian}. Set $$ W = \Spec(A) \times_X \coprod X_i = \coprod \Spec(A) \times_X X_i $$ Since $\coprod X_i \to X$ is universally submersive, there is a specialization $w' \leadsto w$ in $W$ such that $w'$ maps to the generic point of $\Spec(A)$ and $w$ maps to the closed point of $\Spec(A)$. (If not, then the closed fibre of $W \to \Spec(A)$ is stable under generalizations, hence open, which contradicts the fact that $W \to \Spec(A)$ is submersive.) Say $w' \in \Spec(A) \times_X X_i$ so of course $w \in \Spec(A) \times_X X_i$ as well. Let $x'_i \leadsto x_i$ be the image of $w' \leadsto w$ in $X' \times_X X_i$. Since $x'_i \in X'_i$ and since $X'_i \subset X' \times_X X_i$ is a closed subscheme we see that $x_i \in X'_i$. Since $x_i$ maps to $x \in X'$ we conclude that $\coprod X'_i \to X'$ is surjective! In particular $\{X'_i \to X'\}$ is an fppf covering. But an fppf covering is a ph covering (More on Morphisms, Lemma \ref{more-morphisms-lemma-fppf-ph}). Since $X' \to X$ is proper surjective, we conclude that $\{X'_i \to X\}$ is a ph covering and the proof is complete.