I'm not sure if this is a research level question, but:


Let $F:Rep\_A \to Rep\_B$ be an exact cocomplete functor between representation categories of finite dimensional $k$ algebras, where $k$ has charecteristic zero and is algebraically closed. By Eilenberg-Watts $F$ has both left and right adjoint. Are they naturally isomorphic?



Question on Eilenberg-Watts theorem
No. Suppose $F=-\otimes\_AM$, where $M$ is a finite dimensional $B$-$A$-bimodule that is projective as a left $A$-module (so that $F$ is exact).


The right adjoint is $\text{Hom}\_B(M,-)$. But unless $M$ is also projective as a right $B$-module this is not right exact, and so can’t be a left adjoint.


So, for example, take $A=k$ and $M$ any non-projective $B$-module.


