Remark 20.54.5. The map (20.54.2.1) is compatible with the base change map of Remark 20.28.3 in the following sense. Namely, suppose that

is a commutative diagram of ringed spaces. Let $E \in D(\mathcal{O}_ X)$ and $K \in D(\mathcal{O}_ Y)$. Then the diagram

is commutative. Here arrows labeled $t$ are gotten by an application of Lemma 20.27.3, arrows labeled $b$ by an application of Remark 20.28.3, arrows labeled $p$ by an application of (20.54.2.1), and $c$ comes from $L(g')^* \circ Lf^* = L(f')^* \circ Lg^*$. We omit the verification.

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