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

Assume the free functor reflects isomorphisms. If $f,g:X\rightrightarrows Y$ satisfy $Ff=Fg$, let $q:Y\to Q$ be their coequalizer in sets. As a left adjoint, $F$ preserves it. Since the coequalizer of an equal pair is an identity, $Fq$ is an isomorphism. Reflection makes $q$ an isomorphism, so $f=g$. Thus $F$ is faithful.