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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