Lemma 29.33.2. Let $a : X \to S$ and $b : Y \to S$ be morphisms of schemes. Let $\mathcal{F}$ and $\mathcal{F}'$ be quasi-coherent $\mathcal{O}_ X$-modules. Let $D : \mathcal{F} \to \mathcal{F}'$ be a differential operator of order $k$ on $X/S$. Let $\mathcal{G}$ be a quasi-coherent $\mathcal{O}_ Y$-module. Then there is a unique differential operator
\[ D' : \text{pr}_1^*\mathcal{F} \otimes _{\mathcal{O}_{X \times _ S Y}} \text{pr}_2^*\mathcal{G} \longrightarrow \text{pr}_1^*\mathcal{F}' \otimes _{\mathcal{O}_{X \times _ S Y}} \text{pr}_2^*\mathcal{G} \]
of order $k$ on $X \times _ S Y / Y$ such that $ D'(s \otimes t) = D(s) \otimes t $ for local sections $s$ of $\mathcal{F}$ and $t$ of $\mathcal{G}$.
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