I'm working through "Basic Category Theory" by Tom Leinster and am trying to get clarity on how to reason about things... one thing I'm not sure about is how to think about what a functor being faithful and/or full gives us (well, beyond the obvious of being injective and/or surjective in the mapping of morphisms and objects).
So later on there is a question: show that for any adjunction, the right adjoint is full and faithful if and only if the counit is an isomorphism.
So I'm trying to think through the "if the right adjoint is full and faithful, the counit is an isomorphism" case. But I'm not really sure how to think about what being full and faithful gives us. Is it that we know there is a section (axiom of choice!!) and a retract? Is it that there has to be an inverse? (I don't think this is true, right? just because there are separate morphisms that will cancel it out on either side doesn't mean there is one that does both...right?)
So I'm at a loss of how to use full/faithfulness to reason about whether there is an isomorphism, because those properties only related to the right adjoint, not the left.
I'm probably missing something obvious. Thanks for your help.
Update: am trying to do research on this and haven't found a proof, but have found an alternate statement here: http://ncatlab.org/nlab/show/reflective+subcategory. See proposition one.