Definition 36.1:

For $n\geq 1$, the degree of a continuous map $f:S^n\longrightarrow S^n$ is defined to be the integer $m$ such that

$$H_n(f)(\overline{\eta}) = m\;\overline{\eta} \eqno(36.1)$$

where $\overline{\eta}$ is a generator for the infinite cyclic group $H_n(S^n)$. Since $H_n(f): H_n(S^n)\longrightarrow H_n(S^n)$ is a group homomorphism the choice of either of the two generators would yield the same result.