I am reading about Bott Periodicity in Atiyah's "K-Theory". He states a Corollary
$K^{-2}(point) = K^2(S^2)\cong \mathbb{Z}$, generated by the Bott generator $b$.
Then he says that
This corollary amounts to the statement that $\pi_1(U(1))\rightarrow \varinjlim_n \pi_1(U(n))$ is an isomorphism.
I know that since $S^1$ is compact and $U(n)\subseteq U(n+1)$ is closed, $\varinjlim_n[S^1,U(n)]\cong [S^1, U]\cong \tilde{K}(\Sigma S^1)=\tilde{K}(S^2)$. I know $\tilde{K}(S^2) = K^{-2}(point)$ is cyclic generated by $b$.
I also see why the first corollary is true since $K^{-2}(point)\cong K(point)$ by Bott periodicity, then $K(point)\cong K(S^2\times point)\cong K(S^2)\cong K^{2}(S^2)$ by more Bott periodicity.
$\pi_1(U(1))$ is certainly also $\mathbb{Z}$, so I guess I see that they are isomorphic. But what is this map exactly, and how, precisely does it correspond to the Bott map?