Consider the functor $F:\textbf{Set}\rightarrow \textbf{Grp}$ which associates to $s$ the free group generated by $s$, which is (by definition) a left adjoint.
Is there an element free proof of the fact that $F$ is faithful ? Recall that $F$ is faithful iff the unit $s\rightarrow UF s$ is always injective. Is there some general theorem that gives conditions for forgetful functors to have faithful left adjoints ?
Note that the left adjoint to $\textbf{Grp}\rightarrow\textbf{Mon}$ is not faithful so it really depends on the considered forgetful functor. The left adjoint to $\textbf{Ab}\rightarrow \textbf{Grp}$ is also unfaithful.
My motivation comes from the following question: prove that the canonical map (which is an adjunction unit) from a free $n$-category into the generated free $(n,n-1)$-category is injective. I don't think I would manage (though this is probably because I'm unwilling to try...) to write a proper proof using an explicit construction for the generated $(n,n-1)$-category.