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

Let $(A,a)$ be a $T$-algebra with at least two elements, and suppose $Ff:FX\to FY$ is an isomorphism. For every $T$-algebra $A$, precomposition with $Ff$ gives a bijectionBy the free-forgetful adjunction this isFor one set $UA$ with at least two elements, bijectivity of this precomposition forces $f$ to be bijective: surjectivity detects injectivity of $f$, and injectivity detects surjectivity. Thus $f$ is an isomorphism. The free functor reflects isomorphisms, completing the cycle of equivalences.