When are counits monic?

Question

The counit of an adjunction is split monic precisely when the right adjoint is full. Can anything nice be said about when the counit is monic (ideally in terms of the induced comonad)?

Answer 1

As noted in a comment (in the dual case), if the left adjoint is fully faithful then this property has a name: a coreflective subcategory whose counits are mono is called monocoreflective (the dual of "epireflective"). This is of course a property of the induced comonad (since a coreflection is the same as an idempotent comonad). I don't think it has any more concrete characterization, but you can google it and read more about it.

However, we can say that the general case reduces to this special one. Namely, in an arbitrary adjunction if the counits are monic, then since $\epsilon \circ F G \epsilon = \epsilon \circ \epsilon F G$ by naturality, monicity of $\epsilon$ gives $F G \epsilon = \epsilon F G$, and therefore the adjunction is idempotent. Thus it factors as a composite of a reflection $F_1\dashv G_1$ (with invertible counit) and a coreflection $F_2 \dashv G_2$ (with invertible unit), in which case monicity of $\epsilon$ is equivalent to monicity of $\epsilon_2$. Since an adjunction is also idempotent if and only if its induced comonad is idempotent, we can say that for a general adjunction the counit is monic if and only if the induced comonad is idempotent and corresponds to a monocoreflective subcategory.