II.4.4

This has H = 4 (can use II.3.11d, II.3.13, II.2.7), V = ....

HAPPY

 * First show $$f(Z)$$ is separated and of finite type:
 * Recall cor. II.4.8, show the composition $$ Z \to X \to Y$$ is proper.
 * Conclude the natural inclusion $$f(Z) \to Y$$ is a closed immersion, hence of finite type and separated.
 * Conclude that $$f(Z) \to Y \to S$$ is separated.
 * Properness:
 * For any $$Y' \to Y$$ argue $$Y' \times Z \to Y' \times f(Z)$$ is surjective. Can apply ex II.2.7
 * For any closed set $$C \subset Y' \times f(Z)$$, have $$n(C) = (n \circ m)(m^{-1}(C))$$, where $$ Y' \times Z \xrightarrow{m} Y' \times f(Z) \xrightarrow{n} Y'$$