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.

  • $\begingroup$ What do you consider trivial here? $\endgroup$ – Arnaud D. Apr 26 '16 at 21:10
  • $\begingroup$ The identity adjunction, for example. $\endgroup$ – mad_algebraist 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.

I don't know if there are any interesting necessary condition, though.

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  • $\begingroup$ I'm not sure if inclusions of reflective subcategories are full examples per se, since it would seem that only the counit is an isomorphisn (a group isn't always isomorphic to its abelianization, nor a presheaf to its sheafification). However, there are many non-trivial adjoint equivalences---it would be interesting to see an example which is not an equivalence. $\endgroup$ – mad_algebraist Apr 26 '16 at 21:58
  • $\begingroup$ Having only the counit be an isomorphism is enough to have the identities you require. $\endgroup$ – Arnaud D. Apr 26 '16 at 22:02

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