Let $\mathcal{P}$ be a property of morphisms of schemes which satisfies conditions (1), (2) and (3) of the lemma. By Lemma \ref{lemma-precompose-property-local-source} we see that $\mathcal{P}$ is stable under precomposing with \'etale morphisms. By Lemma \ref{lemma-pullback-property-local-target} we see that $\mathcal{P}$ is stable under \'etale base change. Hence it suffices to prove part (3) of Definition \ref{definition-local-source-target} holds. \medskip\noindent More precisely, suppose that $f : X \to Y$ is a morphism of schemes which satisfies Definition \ref{definition-local-source-target} part (3)(b). In other words, for every $x \in X$ there exists an \'etale morphism $a_x : U_x \to X$, a point $u_x \in U_x$ mapping to $x$, an \'etale morphism $b_x : V_x \to Y$, and a morphism $h_x : U_x \to V_x$ such that $f \circ a_x = b_x \circ h_x$ and $h_x$ has $\mathcal{P}$. The proof of the lemma is complete once we show that $f$ has $\mathcal{P}$. Set $U = \coprod U_x$, $a = \coprod a_x$, $V = \coprod V_x$, $b = \coprod b_x$, and $h = \coprod h_x$. We obtain a commutative diagram $$ \xymatrix{ U \ar[d]_a \ar[r]_h & V \ar[d]^b \\ X \ar[r]^f & Y } $$ with $a$, $b$ \'etale, $a$ surjective. Note that $h$ has $\mathcal{P}$ as each $h_x$ does and $\mathcal{P}$ is \'etale local on the target. Because $a$ is surjective and $\mathcal{P}$ is \'etale local on the source, it suffices to prove that $b \circ h$ has $\mathcal{P}$. This reduces the lemma to proving that $\mathcal{P}$ is stable under postcomposing with an \'etale morphism. \medskip\noindent During the rest of the proof we let $f : X \to Y$ be a morphism with property $\mathcal{P}$ and $g : Y \to Z$ is an \'etale morphism. Consider the following statements: \begin{enumerate} \item[(-)] With no additional assumptions $g \circ f$ has property $\mathcal{P}$. \item[(A)] Whenever $Z$ is affine $g \circ f$ has property $\mathcal{P}$. \item[(AA)] Whenever $X$ and $Z$ are affine $g \circ f$ has property $\mathcal{P}$. \item[(AAA)] Whenever $X$, $Y$, and $Z$ are affine $g \circ f$ has property $\mathcal{P}$. \end{enumerate} Once we have proved (-) the proof of the lemma will be complete. \medskip\noindent Claim 1: (AAA) $\Rightarrow$ (AA). Namely, let $f : X \to Y$, $g : Y \to Z$ be as above with $X$, $Z$ affine. As $X$ is affine hence quasi-compact we can find finitely many affine open $Y_i \subset Y$, $i = 1, \ldots, n$ such that $X = \bigcup_{i = 1, \ldots, n} f^{-1}(Y_i)$. Set $X_i = f^{-1}(Y_i)$. By Lemma \ref{lemma-pullback-property-local-target} each of the morphisms $X_i \to Y_i$ has $\mathcal{P}$. Hence $\coprod_{i = 1, \ldots, n} X_i \to \coprod_{i = 1, \ldots, n} Y_i$ has $\mathcal{P}$ as $\mathcal{P}$ is \'etale local on the target. By (AAA) applied to $\coprod_{i = 1, \ldots, n} X_i \to \coprod_{i = 1, \ldots, n} Y_i$ and the \'etale morphism $\coprod_{i = 1, \ldots, n} Y_i \to Z$ we see that $\coprod_{i = 1, \ldots, n} X_i \to Z$ has $\mathcal{P}$. Now $\{\coprod_{i = 1, \ldots, n} X_i \to X\}$ is an \'etale covering, hence as $\mathcal{P}$ is \'etale local on the source we conclude that $X \to Z$ has $\mathcal{P}$ as desired. \medskip\noindent Claim 2: (AAA) $\Rightarrow$ (A). Namely, let $f : X \to Y$, $g : Y \to Z$ be as above with $Z$ affine. Choose an affine open covering $X = \bigcup X_i$. As $\mathcal{P}$ is \'etale local on the source we see that each $f|_{X_i} : X_i \to Y$ has $\mathcal{P}$. By (AA), which follows from (AAA) according to Claim 1, we see that $X_i \to Z$ has $\mathcal{P}$ for each $i$. Since $\{X_i \to X\}$ is an \'etale covering and $\mathcal{P}$ is \'etale local on the source we conclude that $X \to Z$ has $\mathcal{P}$. \medskip\noindent Claim 3: (AAA) $\Rightarrow$ (-). Namely, let $f : X \to Y$, $g : Y \to Z$ be as above. Choose an affine open covering $Z = \bigcup Z_i$. Set $Y_i = g^{-1}(Z_i)$ and $X_i = f^{-1}(Y_i)$. By Lemma \ref{lemma-pullback-property-local-target} each of the morphisms $X_i \to Y_i$ has $\mathcal{P}$. By (A), which follows from (AAA) according to Claim 2, we see that $X_i \to Z_i$ has $\mathcal{P}$ for each $i$. Since $\mathcal{P}$ is local on the target and $X_i = (g \circ f)^{-1}(Z_i)$ we conclude that $X \to Z$ has $\mathcal{P}$. \medskip\noindent Thus to prove the lemma it suffices to prove (AAA). Let $f : X \to Y$ and $g : Y \to Z$ be as above $X, Y, Z$ affine. Note that an \'etale morphism of affines has universally bounded fibres, see Morphisms, Lemma \ref{morphisms-lemma-etale-locally-quasi-finite} and Lemma \ref{morphisms-lemma-locally-quasi-finite-qc-source-universally-bounded}. Hence we can do induction on the integer $n$ bounding the degree of the fibres of $Y \to Z$. See Morphisms, Lemma \ref{morphisms-lemma-etale-universally-bounded} for a description of this integer in the case of an \'etale morphism. If $n = 1$, then $Y \to Z$ is an open immersion, see Lemma \ref{lemma-universally-injective-etale-open-immersion}, and the result follows from assumption (3) of the lemma. Assume $n > 1$. \medskip\noindent Consider the following commutative diagram $$ \xymatrix{ X \times_Z Y \ar[d] \ar[r]_{f_Y} & Y \times_Z Y \ar[d] \ar[r]_-{\text{pr}} & Y \ar[d] \\ X \ar[r]^f & Y \ar[r]^g & Z } $$ Note that we have a decomposition into open and closed subschemes $Y \times_Z Y = \Delta_{Y/Z}(Y) \amalg Y'$, see Morphisms, Lemma \ref{morphisms-lemma-diagonal-unramified-morphism}. As a base change the degrees of the fibres of the second projection $\text{pr} : Y \times_Z Y \to Y$ are bounded by $n$, see Morphisms, Lemma \ref{morphisms-lemma-base-change-universally-bounded}. On the other hand, $\text{pr}|_{\Delta(Y)} : \Delta(Y) \to Y$ is an isomorphism and every fibre has exactly one point. Thus, on applying Morphisms, Lemma \ref{morphisms-lemma-etale-universally-bounded} we conclude the degrees of the fibres of the restriction $\text{pr}|_{Y'} : Y' \to Y$ are bounded by $n - 1$. Set $X' = f_Y^{-1}(Y')$. Picture $$ \xymatrix{ X \amalg X' \ar@{=}[d] \ar[r]_-{f \amalg f'} & \Delta(Y) \amalg Y' \ar@{=}[d] \ar[r] & Y \ar@{=}[d] \\ X \times_Z Y \ar[r]^{f_Y} & Y \times_Z Y \ar[r]^-{\text{pr}} & Y } $$ As $\mathcal{P}$ is \'etale local on the target and hence stable under \'etale base change (see Lemma \ref{lemma-pullback-property-local-target}) we see that $f_Y$ has $\mathcal{P}$. Hence, as $\mathcal{P}$ is \'etale local on the source, $f' = f_Y|_{X'}$ has $\mathcal{P}$. By induction hypothesis we see that $X' \to Y$ has $\mathcal{P}$. As $\mathcal{P}$ is local on the source, and $\{X \to X \times_Z Y, X' \to X \times_Y Z\}$ is an \'etale covering, we conclude that $\text{pr} \circ f_Y$ has $\mathcal{P}$. Note that $g \circ f$ can be viewed as a morphism $g \circ f : X \to g(Y)$. As $\text{pr} \circ f_Y$ is the pullback of $g \circ f : X \to g(Y)$ via the \'etale covering $\{Y \to g(Y)\}$, and as $\mathcal{P}$ is \'etale local on the target, we conclude that $g \circ f : X \to g(Y)$ has property $\mathcal{P}$. Finally, applying assumption (3) of the lemma once more we conclude that $g \circ f : X \to Z$ has property $\mathcal{P}$.