Actuallly, I'm not sure that free pearls as defined are in fact projective; for that matter. I do have some observations on epimorphisms of pearls first.

> **Theorem.** A homomorphism $f\colon (G,S)\to(H,T)$ of pearls is an epimorphism if and only if $f\colon G\to H$ is surjective. The same is true for the category of abelian pearls. However, if $f$ is an epimorphism then it need not be the case that $f(S)=T$. 



> **Proposition.** Let $F_2$ be the free group of rank $2$, with free generators $x$ and $y$. Then $(F_2,\{x\})$ is a free pearl, but is not projective in the category of all pearls. Replacing the free group with its abelianization, we obtain a free-abelian pearl that is not projective in the category of abelian pearls.