I am currently studying adjoint functors, on the book "Advanced Modern Algebra", the author shows by a counterexample that $(U,F)$ is not an adjoint pair of functors,where $U$ and $F$ denote the underlying functor and the free functor, respectively.
The counterexample given by Rotman is:
If $H$ is a finite group with more than one element and $Y$ is a set with more than one element, then ${\rm Hom}_{Sets} (UH,Y)$ has more than one element, but ${\rm Hom}_{Groups} (H,FY)$ has only one element. Therefore, $(U,F)$ is not an adjoint functor.
My doubt:
It is not that straightforward for me to observe that there is only one homomorphism between $H$ and $FY$, so any help would be appreciated!