The Stacks project

Remark 56.3.4. Let $R$ be a ring. Let $A$, $B$, $C$ be $R$-algebras. Let $F : \text{Mod}_ A \to \text{Mod}_ B$ and $F' : \text{Mod}_ B \to \text{Mod}_ C$ be $R$-linear, right exact functors which commute with arbitrary direct sums. If by the equivalence of Lemma 56.3.3 the object $K$ in $\text{Mod}_{A \otimes _ R B}$ corresponds to $F$ and the object $K'$ in $\text{Mod}_{B \otimes _ R C}$ corresponds to $F'$, then $K \otimes _ B K'$ viewed as an object of $\text{Mod}_{A \otimes _ R C}$ corresponds to $F' \circ F$.

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