Assume $F$ is faithful. If $\eta_X(x)=\eta_X(y)$, regard $x,y$ as maps $1\rightrightarrows X$. Their free extensions are $F(x),F(y):F1\rightrightarrows FX$, and the adjunction identifies these with $\eta_Xx,\eta_Xy$. They are equal, so faithfulness gives $x=y$. Hence every unit component $\eta_X:X\to TX$ is monic.


Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 119 / 4 / b / ii / Solution

 Solution (source code)