Looking at the list of adjunctions (in CWM) I strongly get the impression that the components $\eta_x$ of the unit $\eta$ involved are all monomorphisms.
But uptil now I have missed/overlooked any theorem about that, and was unable to find a proof for this conjecture myself.
This all in contrast with co-unit $\varepsilon$ wich can be shown to have epimorphisms as components under the extra condition that the adjoint functor is faithful.
I did manage to prove that faithfulness of the left adjoint functor is a sufficient condition in this context, but is it necessary?
Can it be proved that components of units of adjunctions are monomorphisms?
If not then can you provide a counterexample and/or appropriate extra conditions under which this is the case?