Free abelian groups and free groups

Question

So I learned about both free groups and free abelian groups, their constructions, universal properties etc. It seems that they are completely different concepts, even though they are both free objects in their respective categories. I am still a bit confused about the following: if we start with a free group and adjoin commutative relations between their generators, do we get a free abelian group? In other words, is $$ \langle F(a_1, \dots, a_n) \mid a_ia_j=a_ja_i \text{ for } 1 \leq i,j, \leq n \rangle = \mathbb{Z}^{(a_1, \dots, a_n)}$$ true? Are there some examples that will help clear up my confusion? Thanks.