Lemma 56.3.3. Let $R$ be a ring. Let $A$ and $B$ be $R$-algebras. There is an equivalence of categories between
the category of $R$-linear functors $F : \text{Mod}_ A \to \text{Mod}_ B$ which are right exact and commute with arbitrary direct sums, and
the category $\text{Mod}_{A \otimes _ R B}$.
given by sending $K$ to the functor $F$ in (56.3.2.1).
Proof. Let $F$ be an object of the first category. By Lemma 56.3.1 we may assume $F(M) = M \otimes _ A K$ functorially in $M$ for some $A \otimes _\mathbf {Z} B$-module $K$. The $R$-linearity of $F$ immediately implies that the $A \otimes _\mathbf {Z} B$-module structure on $K$ comes from a (unique) $A \otimes _ R B$-module structure on $K$. Thus we see that sending $K$ to $F$ as in (56.3.2.1) is essentially surjective.
To prove that our functor is fully faithful, we have to show that given $A \otimes _ R B$-modules $K$ and $K'$ any transformation $t : F \to F'$ between the corresponding functors, comes from a unique $\varphi : K \to K'$. Since $K = F(A)$ and $K' = F'(A)$ we can take $\varphi $ to be the value $t_ A : F(A) \to F'(A)$ of $t$ at $A$. This maps is $A \otimes _ R B$-linear by the definition of the $A \otimes B$-module structure on $F(A)$ and $F'(A)$ given in the proof of Lemma 56.3.1. $\square$
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