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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?

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If you are familiar with the relationship between adjunctions and monads, then the term unit will make more sense (and counit is the dual notion). As for your other two questions, generally speaking it is easier to think of an adjunctions in terms of the correspondence on hom-sets but it tends to be cleaner and shorter to prove that something is an adjunction by verifying the conditions in terms of the unit and counit. So, at least at this point, it may assist you if you both read about adjunctions and monads and simply think of the unit/counit form of an adjunction as a very neat way to package all the information in an adjunction into very simply diagrams, which tend to be easy to check. Soon thereafter though you should familiarize yourself with how certain properties of the unit/counit reflect properties of the adjunction. This is standard material and can be found, e.g., in Mac Lane.

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