Remark 20.13.2. Here is a down-to-earth explanation of the meaning of Lemma 20.13.1. It says that given $f : X \to Y$ and $\mathcal{F} \in \textit{Mod}(\mathcal{O}_ X)$ and given an injective resolution $\mathcal{F} \to \mathcal{I}^\bullet$ we have

$\begin{matrix} R\Gamma (X, \mathcal{F}) & \text{is represented by} & \Gamma (X, \mathcal{I}^\bullet ) \\ Rf_*\mathcal{F} & \text{is represented by} & f_*\mathcal{I}^\bullet \\ R\Gamma (Y, Rf_*\mathcal{F}) & \text{is represented by} & \Gamma (Y, f_*\mathcal{I}^\bullet ) \end{matrix}$

the last fact coming from Leray's acyclicity lemma (Derived Categories, Lemma 13.16.7) and Lemma 20.11.10. Finally, it combines this with the trivial observation that

$\Gamma (X, \mathcal{I}^\bullet ) = \Gamma (Y, f_*\mathcal{I}^\bullet ).$

to arrive at the commutativity of the diagram of the lemma.

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