I'm currently reading "CMI Summer School Notes on $p$-adic Hodge Theory"(https://math.stanford.edu/~conrad/papers/notes.pdf) p.51 and could not understand why there exists a canonical algebra structure $$\overline{k} \to \mathcal{O}_{\overline{K}}/(p)$$
where $K$ is a finite extension of $\mathbb{Q}_p$ and $\overline{K}$ is a fixed algebraic closure of $K$, $\mathcal{O}_{\overline{K}}$ is its ring of integers and $\overline{k}$ is its residue field.
Here is my attempt. As $\mathcal{O}_{\overline{K}}/(p)$ is canonically isomorphic to $\mathcal{O}_{\mathbb{C}_p}/(p)$, it suffices to construct a ring morphism $\overline{k} \to \mathcal{O}_{\mathbb{C}_p}/(p)$. For $x \in \overline{k}$, choose a lift $y_0 \in \mathcal{O}_{\overline{K}}$ of $x$ and also choose a sequence $(y_n)$ in $\mathcal{O}_{\overline{K}}$ such that $y_{n+1}^p=y_n$. Then, I thought, $$x \mapsto \lim y_n^{p^{-n}}$$ is the desired map, is it right? If it is right, I don't see why this yields a well-defined map, i.e., I don't really see why the indeterminacy of the choice of lifts $(y_n)$ is killed by modulo $p$.