# Understanding the unit/counit of an adjunction

I'm trying to understand adjoint functors better and I have to admit I'm a little bit confused by the idea of the unit/counit of an adjunction.

I have written out units, counits and their triangle identities for the few common examples of adjoint functors (tensor-hom, free-forgetful...) but that didn't remove my confusion completely.

I've tried to think about my confusion and I think it all boils down to these questions:

1. Where does the name unit/counit come from? Is there a connection with units of algebras/counits of coalgebras?

2. I always see examples of adjunctions between functors written out using hom-sets (and units/counits are there, lurking in the background). Is there an example where adjunction is written more naturally in the unit/counit language than in the hom-set language?

3. Continuing from the question before, it seems I just don't have a good idea of their use. Are there proofs that are easier to do by using units/counits?