Choose a scheme $V'$ and a surjective \'etale morphism $V' \to Y'$. Choose a scheme $U'$ and a surjective \'etale morphism $U' \to X' \times_{Y'} V'$. Set $V = Y \times_{Y'} V'$ and $U = X \times_{X'} U'$. Then for \'etale local properties of morphisms we can reduce to the morphism of thickenings of schemes $(U \subset U') \to (V \subset V')$ and apply More on Morphisms, Lemma \ref{more-morphisms-lemma-properties-that-extend-over-thickenings}. This proves (1), (4), and (5). \medskip\noindent The properties in (2), (3), (6), (7), and (8) are stable under base change, hence if $f'$ has property $\mathcal{P}$, then so does $f$. See Spaces, Lemma \ref{spaces-lemma-base-change-immersions}, and Morphisms of Spaces, Lemmas \ref{spaces-morphisms-lemma-base-change-proper}, \ref{spaces-morphisms-lemma-base-change-integral}, and \ref{spaces-morphisms-lemma-base-change-monomorphism}. Hence in each case we need only to prove that if $f$ has the desired property, so does $f'$. \medskip\noindent Case (2) follows from case (5) of Lemma \ref{lemma-thicken-property-morphisms} and the fact that the finite morphisms are precisely the integral morphisms that are locally of finite type (Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-finite-integral}). \medskip\noindent Case (3). This follows immediately from Limits of Spaces, Lemma \ref{spaces-limits-lemma-check-closed-infinitesimally}. \medskip\noindent Proof of (6). As $f$ is a monomorphism we have $X = X \times_Y X$. We may apply the results proved so far to the morphism of thickenings $(X \subset X') \to (X \times_Y X \subset X' \times_{Y'} X')$. We conclude $\Delta_{X'/Y'} : X' \to X' \times_{Y'} X'$ is a closed immersion by (3). In fact $\Delta_{X'/Y'}$ induces a bijection $|X'| \to |X' \times_{Y'} X'|$, hence $\Delta_{X'/Y'}$ is a thickening. On the other hand $\Delta_{X'/Y'}$ is locally of finite presentation by Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-diagonal-morphism-finite-type}. In other words, $\Delta_{X'/Y'}(X')$ is cut out by a quasi-coherent sheaf of ideals $\mathcal{J} \subset \mathcal{O}_{X' \times_{Y'} X'}$ of finite type. Since $\Omega_{X'/Y'} = 0$ by (5) we see that the conormal sheaf of $X' \to X' \times_{Y'} X'$ is zero. (The conormal sheaf of the closed immersion $\Delta_{X'/Y'}$ is equal to $\Omega_{X'/Y'}$; this is the analogue of Morphisms, Lemma \ref{morphisms-lemma-differentials-diagonal} for algebraic spaces and follows either by \'etale localization or by combining Lemmas \ref{lemma-differentials-relative-immersion-section} and \ref{lemma-differential-product}; some details omitted.) In other words, $\mathcal{J}/\mathcal{J}^2 = 0$. This implies $\Delta_{X'/Y'}$ is an isomorphism, for example by Algebra, Lemma \ref{algebra-lemma-ideal-is-squared-union-connected}. \medskip\noindent Proof of (7). If $f : X \to Y$ is an immersion, then it factors as $X \to V \to Y$ where $V \to Y$ is an open subspace and $X \to V$ is a closed immersion, see Morphisms of Spaces, Remark \ref{spaces-morphisms-remark-immersion}. Let $V' \subset Y'$ be the open subspace whose underlying topological space $|V'|$ is the same as $|V| \subset |Y| = |Y'|$. Then $X' \to Y'$ factors through $V'$ and we conclude that $X' \to V'$ is a closed immersion by part (3). \medskip\noindent Case (8) follows from Lemma \ref{lemma-thicken-property-morphisms} and the definition of proper morphisms as being the quasi-compact, universally closed, and separated morphisms that are locally of finite type.