Proof.
We write everything out completely.
Proof of (1). Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Let $\{ X_ i \to X\} _{i \in I}$ be an étale covering of $X$. If each composition $h_ i : X_ i \to Y$ has $\mathcal{P}$, then for each $|x| \in X$ we can find an $i \in I$ and a point $x_ i \in |X_ i|$ mapping to $x$. Then $(X_ i, x_ i) \to (X, x)$ is an étale morphism of pairs, and $\text{id}_ Y : Y \to Y$ is a smooth morphism, and $h_ i$ is as in part (3) of Definition 74.21.1. Thus we see that $f$ has $\mathcal{P}$. Conversely, if $f$ has $\mathcal{P}$ then each $X_ i \to Y$ has $\mathcal{P}$ by Definition 74.21.1 part (1).
Proof of (2). Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Let $\{ Y_ i \to Y\} _{i \in I}$ be a smooth covering of $Y$. Write $X_ i = Y_ i \times _ Y X$ and $h_ i : X_ i \to Y_ i$ for the base change of $f$. If each $h_ i : X_ i \to Y_ i$ has $\mathcal{P}$, then for each $x \in |X|$ we pick an $i \in I$ and a point $x_ i \in |X_ i|$ mapping to $x$. Then $X_ i \to X \times _ Y Y_ i$ is an étale morphism (because it is an isomorphism), $Y_ i \to Y$ is smooth, and $h_ i$ is as in part (3) of Definition 74.20.1. Thus we see that $f$ has $\mathcal{P}$. Conversely, if $f$ has $\mathcal{P}$, then each $X_ i \to Y_ i$ has $\mathcal{P}$ by Definition 74.20.1 part (2).
Proof of (3). Assume $f : X \to Y$ has $\mathcal{P}$ and $g : Y \to Z$ is étale. The morphism $X \to Y \times _ Z X$ is étale as a morphism between algebraic spaces étale over $X$ ( Properties of Spaces, Lemma 66.16.6). Also $Y \to Z$ is étale hence a smooth morphism. Thus the diagram
\[ \xymatrix{ X \ar[d] \ar[r]_ f & Y \ar[d] \\ X \ar[r]^{g \circ f} & Z } \]
works for every $x \in |X|$ in part (3) of Definition 74.20.1 and we conclude that $g \circ f$ has $\mathcal{P}$.
Proof of (4). Let $f : X \to Y$ be a morphism and $g : Y \to Z$ étale such that $g \circ f$ has $\mathcal{P}$. Then by Definition 74.21.1 part (2) we see that $\text{pr}_ Y : Y \times _ Z X \to Y$ has $\mathcal{P}$. But the morphism $(f, 1) : X \to Y \times _ Z X$ is étale as a section to the étale projection $\text{pr}_ X : Y \times _ Z X \to X$, see Morphisms of Spaces, Lemma 67.39.11. Hence $f = \text{pr}_ Y \circ (f, 1)$ has $\mathcal{P}$ by Definition 74.21.1 part (1).
$\square$
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