Let $K$ be a local field and let $\widehat{K^*}$ and $\widehat{\mathbb{Z}}$ denote the profinite completions of $K$ and $\mathbb{Z}$. As the title suggests I'm having difficulties understanding the isomorphism ${\mathcal{O}^*_K} \times \widehat{\mathbb{Z}} \cong \widehat{K^*}$ that is given by $(u, z) \mapsto u \pi^z$ where $\pi$ is a uniformizer of ${\mathcal{O}^*_K}$. Concerning this, I have three questions:
How can we view $\pi^z$ as an element of $\widehat{K^*}$?
What does this map look like levelwise when derived as the map coming from the universal property of direct limits?
Why is $\widehat{\mathcal{O}_K^*} = \mathcal{O}_K^*?$