# Existence of adjoints with commutativity condition

Are there (non-trivial) examples of adjunctions $F \operatorname{\dashv} G$ with unit and counit $\eta$, $\epsilon$ such that $$\eta GF = GF \eta$$ and $$FG \epsilon = \epsilon FG,$$ and if so, what are necessary or sufficient conditions for this to happen?

For example, this fails for the product-hom adjunction in the category of sets.

• What do you consider trivial here? Apr 26 '16 at 21:10
• The identity adjunction, for example. Apr 26 '16 at 21:23

The triangle identities imply that $$G\epsilon F \circ GF\eta=id_{GF} =G\epsilon F\circ \eta GF$$ and $$FG\epsilon \circ F\eta G= id_{FG}=\epsilon FG\circ F\eta G.$$ Thus $FG\epsilon=\epsilon FG$ and $GF\eta =\eta GF$ as soon as $\epsilon$ or $\eta$ is an isomorphism. Since $\epsilon$ (resp. $\eta$) is an isomorphism if and only if $G$ (resp. $F$) is fully faithful, there are a lot of examples, for example the classical adjunction between groups and abelian groups.
In fact it is enough to have $\epsilon_X$ be a monomorphism for all $X$ (or dually, $\eta_Y$ an epimorphism for all $Y$). Indeed, if it is the case, since $G$ preserves monomorphisms $G\epsilon_X$ is a monomorphism, and the identity $G\epsilon\circ \eta G=id_{G}$ implies that $G\epsilon$ is an isomorphism. It is thus enough to have $G$ full or $F$ faithful.