Unit and counit

Summary

The unit and the counit of an adjunction $F \dashv G$ are the two natural transformations $\eta \colon \mathrm{Id} \Rightarrow G F$ and $\varepsilon \colon F G \Rightarrow \mathrm{Id}$ that it determines. They satisfy the two triangle identities, and conversely they determine the adjunction. The unit and the counit give the monad and the comonad of the adjunction.