Since I'm working in the niche category of $C^\infty$-rings, I'll state my question in general terms, though if you know more about this category, feel free to give a more specific answer.
I'm given an surjective hom $r:A\to B$, where $A$ is a quotient of a free object. I want to show that $r$ is a retraction. My first thought is that we can view $B=F/J$ as a quotient of the same free object $F$ and define an inverse on said free object. Then what remains to be shown is, that the kernel of this morphism is containing $J$. Here I fall flat. I am happy about any new thoughts or general ideas and feedback