Remark 103.11.3. Let $f : \mathcal{X} \to \mathcal{Y}$ be a quasi-compact and quasi-separated morphism of algebraic stacks. Let $\mathcal{F}$ and $\mathcal{G}$ be in $\mathit{QCoh}(\mathcal{O}_\mathcal {X})$. Then there is a canonical commutative diagram

The vertical arrow $c$ on the right is the naive relative cup product (in degree $0$), see Cohomology on Sites, Section 21.33. The source and target of $c$ are in $\textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {X})$, see Proposition 103.8.1. Applying $Q$ to $c$ we obtain the left vertical arrow as $Q$ commutes with tensor products, see Remark 103.10.6. This construction is functorial in $\mathcal{F}$ and $\mathcal{G}$.

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