Homomorphism condition for a subset of a group generates the whole group

Question

I wonder whether the following statement is true:

Let F is a group and X is a subset of F. Then

$\left< X \right> =F\quad \Longleftrightarrow \quad$ For any group G and any function $\phi :X\rightarrow G$ , homomorphic extension ${ \phi }^{ * }:F\rightarrow G $ of $\phi$ is unique if such homomorphic extension exist.

($\Rightarrow $) is obvious but the converse is difficult for me.

I don't know about the notion of universal property, and the question is derived from formal definition of free group.

Answer 1

Yes, this is true. Suppose that $H = \langle X \rangle$ is a proper subgroup of $F$, and let $G = F *_H F$ be the free product of two copies of $F$ amalgamated in their common subgroup $H$. Then the embedding $i:H \to G$ can be extended to $F$ in two different ways, in which the images are the two factors in the amalgamated product.