Composite of a finite morphism and a proper morphism is proper

Finite type and quasi-compactness of $g\circ h$. Fix a point $x\in X$. Choose an affine open $\mathrm{Spec}B\subseteq Y$ with $h(x)\in \mathrm{Spec}B$ and an affine open $\mathrm{Spec}A\subseteq S$ with $g(\mathrm{Spec}B)\subseteq \mathrm{Spec}A$ and $A\to B$ of finite type, as [F2] permits for the finite-type morphism $g$ [F5]; write $h^{-1}(\mathrm{Spec}B)=\mathrm{Spec}C$, so that $C$ is module-finite over $B$ [F1]. A module generating set of $C$ over $B$ generates $C$ as a $B$-algebra, so $B\to C$ is of finite type, and a generating set of $B$ over $A$ together with one of $C$ over $B$ generates $C$ over $A$, so $C$ is of finite type over $A$; the affine chart $\mathrm{Spec}C$ of $x$ over $\mathrm{Spec}A$ therefore witnesses that $g\circ h$ is locally of finite type [F2]. It is quasi-compact as well: $g$ is of finite type, hence quasi-compact, by [F5] and [F2], and the finite morphism $h$ is proper by [F9], hence of finite type and quasi-compact, again by [F5] and [F2]; for a quasi-compact open $U\subseteq S$ the preimage $(g\circ h{)}^{-1}(U)=h^{-1}(g^{-1}(U))$ is then the preimage under the quasi-compact morphism $h$ of the quasi-compact open $g^{-1}(U)$ [F2]. Hence $g\circ h$ is of finite type.

Universal closedness. Base change along any $T\to S$ gives $(g\circ h{)}_T=g_T\circ h_{Y_T}$ where $h_{Y_T}:X_T\to Y_T$ is the base change of $h$ and is again finite, since finiteness is checked on affine charts and tensor products of module-finite algebras are module-finite [F1]; by [F6] it is universally closed, and $g_T$ is closed because $g$ is universally closed [F4]. A composite of closed maps is closed, so $(g\circ h{)}_T$ is closed for every $T$, i.e. $g\circ h$ is universally closed [F4].

Separatedness. The finite morphism $h$ is proper by [F9], hence separated by [F5], so its diagonal ${\mathrm{\Delta }}_h:X\to X{\times }_{Y}X$ is a closed immersion, and for $g$ separated ${\mathrm{\Delta }}_g:Y\to Y{\times }_{S}Y$ is one too [F3]. The diagonal of $g\circ h$ factors canonically as $X\stackrel{{\mathrm{\Delta }}_h}{\to }X{\times }_{Y}X\stackrel{i}{\to }X{\times }_{S}X,$ because both composites with the two projections to $X$ are the identity; here $i$ is the natural inclusion, the base change of ${\mathrm{\Delta }}_g$ along $h{\times }_{S}h:X{\times }_{S}X\to Y{\times }_{S}Y$ and is therefore a closed immersion [F7]. A composite of closed immersions is a closed immersion — closedness of the image is preserved by composition of homeomorphisms onto closed subsets, and surjectivity of structure sheaves is stable under composition — so ${\mathrm{\Delta }}_{g\circ h}$ is a closed immersion and $g\circ h$ is separated [F3]; since it is of finite type by step 1.1, it is quasi-separated [F3]. The separatedness of $h$ used above is thus a consequence of finiteness, via [F9] and [F5].

Valuative criterion. Let $R$ be a valuation ring with fraction field $K$ and let a valuative diagram for $g\circ h$ be given, with generic map $\mathrm{Spec}K\to X$ and base map $\mathrm{Spec}R\to S$. Composing the generic map with $h$ yields a valuative diagram for $g$, which by [F8] and the properness of $g$ has exactly one lift $v:\mathrm{Spec}R\to Y$ under Choice [F10]. Now $v$ and the generic map form a valuative diagram for $h$; the finite morphism $h$ is proper by [F9], hence of finite type and separated by [F5] and quasi-separated by [F3], so [F8] applied to $h$ gives a lift $u:\mathrm{Spec}R\to X$ with $h\circ u=v$, which is a lift of the original diagram. For uniqueness let $u_1,u_2$ be two lifts of that diagram; then $h\circ u_1$ and $h\circ u_2$ are two lifts of the induced diagram for $g$, so $h\circ u_1=h\circ u_2$ by uniqueness for $g$, and $u_1,u_2$ are then two lifts of one valuative diagram for $h$, so $u_1=u_2$ by uniqueness for the proper morphism $h$ [F9]. Hence every valuative diagram for $g\circ h$ has exactly one lift, and since $g\circ h$ is of finite type and quasi-separated by steps 1.1 and 2.1, [F8] gives that $g\circ h$ is proper.

By step 1.1 the composite $g\circ h$ is of finite type and by step 2.1 it is quasi-separated; by step 2.1 it is separated; by step 1.2 it is universally closed; and by step 3.1 it satisfies the valuative criterion under Choice. The criterion [F8] applied in its sufficient direction gives that $g\circ h$ is proper, which is also the conjunction of separated, finite type and universally closed by [F5]. Choice is inherited through [F9] in steps 1.1, 2.1 and 3.1, through [F6] in step 1.2, and through [F8] in step 3.1 and this conclusion.