Lemma 56.3.9. Let $R$ be a ring. Let $A$ and $B$ be $R$-algebras. If
is an $R$-linear equivalence of categories, then there exists an isomorphism $A \to B$ of $R$-algebras and an invertible $B$-module $L$ such that $F$ is isomorphic to the functor $M \mapsto (M \otimes _ A B) \otimes _ B L$.