Aguasome functors of modules

Question

Let $A$ and $B$ be commutative rings, $A-\mathbf{Mod}_f$ be the category of finite type $A$-modules. A functor $F:A-\mathbf{Mod}_f\rightarrow B-\mathbf{Mod}_f$ is said to be aguasome if is additive and there is a functorial isomorphism $F(Hom_A(M,N))\simeq Hom_{B}(F(M),F(N))$. Is there a classification of such functors? (à la EILENBERG-WATTS), in particular is every such functor the extension of scalars with respect to a flat morphism?

Answer 1

To answer the second question, there are examples that are not extension of scalars with respect to a flat morphism.

For example, let $\varphi:B\to A$ be a ring epimorphism, and $F$ the obvious functor given by restriction along $\varphi$.

Or you can compose one of the examples you've thought of with one of mine. For example, the functor that takes a $\mathbb{F}_2$-vector space $V$ to the $\mathbb{Z}[t]$-module $V\otimes_\mathbb{Z}\mathbb{Z}[t]$.