# Question involving Watts theorem

Lets $$A$$ and $$B$$ two algebras, $$_AU_B$$ and $$_B V_A$$ two bi-modules such that there exists two isomorphisms $$U \otimes_B V \simeq A$$ and $$V \otimes_A U \simeq B$$. Let $$F: Mod A \to Mod B$$ a functor, if $$F \simeq -\otimes_A U$$ then $$F$$ is an equivalence between $$Mod A$$ and $$Mod B$$.

There are at least two ways to solve this exercise, the first one is showing that $$F$$ is faithfull, full and dense. The another way is showing that there exists a functor $$G: Mod B \to Mod A$$ s.t $$FG \simeq 1_{mod B}$$ and $$1_{Mod A} \simeq GF$$.

Using $$F \simeq -\otimes_A U$$, follows from Watts Theorem that $$F$$ admit a right adjoint functor $$G$$, further from that I've proved that there exists a morphism between $$FG$$ and $$1_{Mod B}$$, but I'm stuck here.

I need some help...

Simply define $$G$$ to be $$-\otimes_B V$$. Then $$GF\simeq -\otimes_A U\otimes_B V$$ and $$FG\simeq-\otimes_B V\otimes_A U$$ are isomorphic to the identity functors since $$U \otimes_B V \simeq A$$ and $$V \otimes_A U \simeq B$$.