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Suppose I consider a locally free coherent sheaf $E$ on $X$ and similarly $F$ on $Y$. Now suppose that I have a global section $s$ of $E \boxtimes F$. Why does this section induce a map $H^0(X,E)^\vee \to H^0(Y,F)$?

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I think I figured it out. It follows from the Kunneth formula, so the global section $s$ lives in $s \in H^0(X\times Y, E \boxtimes F) \cong H^0(X,E) \otimes H^0(Y,F)$. And an element $s \in H^0(X,E) \otimes H^0(Y,F)$ is the same as a map $s \colon H^0(X,E)^\vee \to H^0(Y,F)$.

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