= Solution
The composite of two <finite type morphisms> is of <finite type>: on suitable <affine open subsets>, take the algebra generators of the first algebra together with the images of the algebra generators of the second. To check <separatedness>, factor the <diagonal morphism> of the composite as
$$
X\xrightarrow{\Delta_f}X\times_YX\longrightarrow X\times_ZX.
$$
The first arrow is a <closed immersion>, and the second is the <base change> of the <closed immersion> $\Delta_g:Y\to Y\times_ZY$. Hence their composite is a <closed immersion>.
Finally, after any <base change> $Z'\to Z$, the composite is $X'\xrightarrow{f'}Y'\xrightarrow{g'}Z'$. Both arrows take closed subsets to closed subsets, so their composite does as well. This is <universal closedness>. All three conditions hold, and therefore
$$
\boxed{f,g\text{ proper}\ \Longrightarrow\ gf\text{ proper}.}
$$
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