Vrftjkry

I am trying to understand the basic theory of free abelian groups and let me ask you the following question. This is the excerpt of the book which I'm reading.

Lemma (Universal property of free abelian group): Let $X$- theset of generators of abelian group $F$, $|X|=n$. Then the following
statements are equivalent:

(i) $F=F^{ab}_n$ - free abelian groups and $X$- the set of its
generators.

(ii) Any mapping $varphi:Xto A$ where $A$ - abelian group
induces the homomorphism $tilde{varphi}:Fto A$.

Corollary: Any abelian group with $n$ generators is homomorphic image of some free abelian group $F^{ab}_n$.

Proof: Follows immediately from univeral property.

However, for me is not clear how it follows from lemma. But I have done in the following way: suppose that $(A,+)$ is abelian group with generators ${a_1,a_2,dots, a_n}$. Let $F^{ab}_n$ be some free abelian group with generators $X={x_1,dots,x_n}$. Consider the mapping $varphi:Xto A$ by $varphi: x_imapsto a_i$ then we can consider the homomorphism $tilde varphi: F^{ab}_nto A$ defined via $$tilde varphi:sum k_ix_i mapsto sum k_ia_i.$$

Then easy to see that $tilde varphi(F^{ab}_n)=A$.

Is my proof looks correct?

But I was not able to derive it immediately from Universal property.

Would be very grateful for help!

You are sort of partially reproving the universal property for this case, which is not needed.

The universal property ought to be stated as

For every function $phi: X to A$ where $A$ is an Abelian group, there is a unique homomorphism $f:F= F^{textrm{ab}}_n to A$ such that $f| X=phi$.

Now, if $A$ is Abelian group with your $n$ generators, indeed define $phi(x_i) =a_i$ when ${x_1,ldots,x_n} = X$. Then by the universal property, a homomorphism $f: F to A$ exists, extending $phi$.
As $f[X]=phi[X]$ is a set of generators for $A$, $f[F]=A$, as $f[F]$ is a subgroup containing the $a_i$, and the smallest subgroup containing the $a_i$ is $A$ (that is what it means to be a generating set). So surjectivity is almost automatic, and we're done proving the corollary.