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
\xymatrix{ f_{\mathit{QCoh}, *}\mathcal{F} \otimes _{\mathcal{O}_\mathcal {Y}} f_{\mathit{QCoh}, *}\mathcal{G} \ar[r] \ar[d] & f_*\mathcal{F} \otimes _{\mathcal{O}_\mathcal {Y}} f_*\mathcal{G} \ar[d]^ c \\ f_{\mathit{QCoh}, *}(\mathcal{F} \otimes _{\mathcal{O}_\mathcal {X}} \mathcal{G}) \ar[r] & f_*(\mathcal{F} \otimes _{\mathcal{O}_\mathcal {X}} \mathcal{G}) }
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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