Let $k$ be an algebraically closed field of characteristic $0$, $K$ a function field over $k$, $R=(R,m_R)$ a discrete valuation ring of rank one with residue field $\kappa_R=R/m_R$ such that $k \subset R \subset K$ and the fraction field of $R$ is $K$.
Let $\widehat{R}_{m_R}$ be the completion of $R$ with respect $m_R$. How to show that $\kappa_R$ can be embedded in $\widehat{R}_m$ as $k$-algebra? Is this embedding canonical?
Note that since $ \widehat{R}_{m_R} $ is projective limit of the $R/m^i$ it might be sufficient to construct a compatible family of maps from $\kappa_R$ to $R/m^i$ and check that the induced map to $\widehat{R}_{m_R}$ is injective.
The question is motivated by the problem here to show that the fraction field of $ \widehat{R}_{m_R} $ is isomorphic to $\kappa_R((T))$?
I assume it is known to you that the natural map of a valuation ring's residue field to the residue field of its completion is an isomorphism.
From there, you might use the following, which deserves to be known more widely:
Let $(A, \mathfrak m)$ be a local ring which is complete and Hausdorff (i.e. $A \simeq \projlim A/\mathfrak m^n$). Let the residue field $k_A:=A/\mathfrak m$ be of characteristic $0$. Let $(x_\lambda)$ be any family of elements of $A$ such that their residues mod $\mathfrak m$ form a transcendence basis of $k_A$ over $\mathbb Q$. Then there exists a unique subfield ("field of representatives") $L \subset A$ that contains all $x_\lambda$ and such that $A = L + \mathfrak m$.
See Bourbaki, Algèbre Commutative, ch. IX § 3 no. 3 Théorème 1. Zorn's Lemma is used in the proof. (Similar but more intricate statements -- one needs $p$-bases instead of general transcendence bases -- hold in the case of equal positive characteristic, but are more intricate to prove.)