I am trying to prove that group P is free if and only if it satisfies the following:
for any groups $A,B$, any homomorphism $\pi: P \rightarrow B$ and any epimorphism $\alpha: A \rightarrow B$ there exists homomorphism $\phi: P \rightarrow A$ that $\alpha \phi=\pi$.
I have proven that any free group satisfies that but I can't prove another inclusion.