Lifting residue-field embeddings of unramified extensions

For an unramified extension $L/K$ of non-Archimedean local fields and any finite extension $M/K$, reduction is a bijection between $K$-field homomorphisms $L\to M$ and $k_K$-field homomorphisms $k_L\to k_M$. Choose an integral generator with separable irreducible reduction. Each specified residue image lifts uniquely to a root in $M$ by Hensel lemma. Conversely a $K$-embedding preserves the unique extended valuation and therefore induces the residue embedding. The same argument applies when $M$ is any non-Archimedean local extension with the given compatible embedding of $K$.