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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