A local homomorphism of local rings $(A,\mathfrak{m})\stackrel{\varphi}{\longrightarrow}(B,\mathfrak{n})$ is called a *scalar extension* (terminology due to Hans Schoutens) if: 

 - $\varphi(\mathfrak{m})B=\mathfrak{n}$, and
 - $\varphi$ is a flat extension.

**Theorem.** (Grothendieck, EGA III, Proposition 10.3.1, page 20). *Let $(A,\mathfrak{m})$ be Noetherian local ring with residue field $k$, and let $K$ be a field extension of $k$. Then there exists a scalar extension $$(A,\mathfrak{m})\stackrel{\varphi}{\longrightarrow}(B,\mathfrak{n})$$ from $A$ to a Noetherian local ring $B$, with the property that $B/\mathfrak{n}$ is $k$-isomorphic to $K$.* 

**Note.** Other constructions of scalar extensions (not necessarily with this name) have appeared in 

 - Hochster and Huneke, *$F$-regularity, test elements, and smooth base change*, Trans. Amer. Math. Soc., **346** (1994) (see pages 18-19), and

 - Schoutens, *Classifying singularities up to analytic extensions of scalars*, Ann. of Pure and Applied Logic, **162**, (2011) (also available on the Arxiv, see pages 5-8).

**Question.** *Is scalar extension of local rings a functor?* To be more precise, suppose $A_1\stackrel{\psi}{\longrightarrow} A_2$ is a local homomorphism of Noetherian local rings with residue fields $k_1$ and $k_2$, and let $K$ be a common field extension of $k_1$ and $k_2$. Let $A_1\longrightarrow B_1$ and $A_2\longrightarrow B_2$ be two corresponding scalar extensions. Does $\psi$ extend to a local homomorphism $B_1\longrightarrow B_2$?

If you know of any reference where this question is discussed, please let me know.

**EDIT:** $B_1$ and $B_2$ are *not* any two arbitrary scalar extensions in my question. They are *the* scalar extensions obtained by the method described in the second reference by Schoutens.