Definition 12.2 (Degree of a map):

Suppose that $f : S^1 \longrightarrow S^1$ is a continuous map such that $f(1) = 1$, the degree of $f$ is defined to be the degree of the loop

$$f\circ\eta : t\mapsto f(\exp(\pm 2\pi it)),\quad 0 \leq t \leq 1. \eqno(12.5)$$