Exercise 1, § 3

Let X be a topological space, let $\sigma : X\times \mathbf{I}\rightarrow X$ be a homotopy whose origin and end are equal to the identity mapping of X. Let $x\in X$ and let $\delta \in \pi_1(X, x)$ be the class of the loop $t\mapsto \sigma (x, t)$ at $x$. Prove that $\delta$ belongs to the center of the group $\pi_1(X, x)$.