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Sadly I don't follow his explanation there. Specifically, I mean the fact that $f^!(F) \cong f^*(F) \otimes f^!(\mathcal{O}_Y)$, for $F$ with proper support. As far as the spanning set is concerned, I only know a proof that they are in fact one when the scheme is smooth and projective. In the general case $Y$ is only quasi-projective, I think.
