Past exam of the mathematics course of the University of Cambridge / 2003 / iii / Paper 18 / 5 / ii / Solution

Apply the inverse transpose formula to ${\eta }_A$, which by definition is the transpose of $1_{FA}$. This gives ${\epsilon }_{FA}F{\eta }_A=1_{FA}$. Likewise ${\epsilon }_B$ transposes to $1_{GB}$, so $G{\epsilon }_B{\eta }_{GB}=1_{GB}$. Hence the triangle identities for an adjunction areThese are equalities of natural transformations; their components are exactly the two identities just proved.