Let $V$ be a vector space over $F$, and $S$ a non-empty subset of $V$. We say that $S$ generates $V$ if every $v\in V$ can be written as finite $F$-linear combination of elements of $S$.
I want to express the above definition of generating set in terms of universal property. My question is whether the following is the universal property of $S$ (generating set) and how to prove that it is equivalent to above definition? Just hint is sufficient, I will try to write complete proof.
We say that $S$ generates $V$ if the following holds: any map $\varphi$ from $S$ into any vector space $W$ extends to a linear map $\varphi_1$ from $V$ into/onto $W$.
The universal property you wrote (leaving out any assertion about $\varphi_1$ being injective or surjective) is equivalent to $S$ being linearly independent, not generating $V$. You want to say instead that any $\varphi$ extends to at most one linear map $\varphi_1$ (again, with no condition that $\varphi_1$ is injective or surjective). The hard direction of the proof is that if $S$ has this property, then $S$ generates $V$. To prove this direction, you need to give an example of a $\varphi$ for which $\varphi_1$ is not unique, assuming $S$ does not generate $V$. To do this, let $U$ be the subspace generated by $S$, $W=V/U$, and $\varphi=0$.