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

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 ${\eta }_{X}x,{\eta }_{X}y$. They are equal, so faithfulness gives $x=y$. Hence every unit component ${\eta }_X:X\to TX$ is monic.