The Stacks project

Proof. Proof of (1). Let $\mathcal{J}^\bullet $ be a K-injective complex of $\mathcal{O}_ f$-modules representing $N$. By Cohomology on Sites, Lemma 21.20.10 it follows that $\mathcal{J}^\bullet $ is a K-injective complex of $\mathcal{O}$-modules as well. Let $\mathcal{F}^\bullet $ be a complex of $\mathcal{O}_ f$-modules representing $L$. Then

by Modules on Sites, Lemma 18.11.4 because $\mathcal{J}^\bullet $ is a K-injective complex of $\mathcal{O}$ and of $\mathcal{O}_ f$-modules.

Proof of (2). Let $\mathcal{I}^\bullet $ be a K-injective complex of $\mathcal{O}$-modules representing $K$. Then $R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O}(\mathcal{O}_ f, K)$ is represented by $\mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O}(\mathcal{O}_ f, \mathcal{I}^\bullet )$ which is a K-injective complex of $\mathcal{O}_ f$-modules and of $\mathcal{O}$-modules by Cohomology on Sites, Lemmas 21.20.11 and 21.20.10. Let $\mathcal{F}^\bullet $ be a complex of $\mathcal{O}_ f$-modules representing $L$. Then

by Modules on Sites, Lemma 18.27.8 and because $\mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O}(\mathcal{O}_ f, \mathcal{I}^\bullet )$ is a K-injective complex of $\mathcal{O}_ f$-modules.

Proof of (3). This follows from the fact that $R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O}(\mathcal{O}_ g, \mathcal{I}^\bullet )$ is K-injective as a complex of $\mathcal{O}$-modules and the fact that $\mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O}(\mathcal{O}_ f, \mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O}(\mathcal{O}_ g, \mathcal{H})) = \mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O}(\mathcal{O}_{gf}, \mathcal{H})$ for all sheaves of $\mathcal{O}$-modules $\mathcal{H}$. $\square$