Definition 23.3 (free groups):

The coproduct in the category Gr (known as the free product) of $k$ copies of $\mathbb{Z}$ is called the free group on $k$ generators.

We shall denote a free group on $k$ generators by $F_k$ or if there is a need to specify the generators $a_1, a_2,\dots, a_k$ we shall use the notation $F[a_1, a_2, \dots, a_k]$.