Let $p$ be a prime number, $\mathcal{O}$ the integers of a finite extension of $\mathbb{Q}_p$ with residue field $k$. Let $\mathcal{C}$ be the category of complete, local, noetherian $\mathcal{O}$-algebras with residue field $k$ and with local homomorphisms (a morphism $A \to B$ in $\mathcal{C}$ induces the identity map $k \to k$ on residue fields).

Denote by $\mathcal{D}$ the category of complete, local $\mathcal{O}$-algebras with residue field $k$ which are inverse limit of objects in $\mathcal{C}$ and with local homomorphisms.

If $F$ and $G$ are functors from $\mathcal{C}$ or $\mathcal{D}$ to $Sets$ the category of sets, we say that morphism of functors $F \to G$ is formally smooth if for every surjection $B \to A$ in $\mathcal{C}$ or $\mathcal{D}$, the induced map $F(B) \to F(A) \times_{G(A)}G(B)$ is surjective.

We can show that if $R\to S$ is a morphism in $\mathcal{C}$, then the morphism of functors $Hom(S,-) \to Hom(R,-)$is formally smooth if and only if $S$ is a power series ring over $R$ (see for example http://www.ams.org/journals/tran/1968-130-02/S0002-9947-1968-0217093-3/home.html Proposition 2.5).

Is the same statement true if we work in $\mathcal{D}$ instead of $\mathcal{C}$, allowing power series with an infinite number of variables  ?

Let $p$ be a prime number, $\mathcal{O}$ the integers of a finite extension of $\mathbb{Q}_p$ with residue field $k$. Let $\mathcal{C}$ be the category of complete, local, noetherian $\mathcal{O}$-algebras with residue field $k$ and with local homomorphisms inducing identity map $k \to k$ on residue fields.

Denote by $\mathcal{D}$ the category of complete, local $\mathcal{O}$-algebras with residue field $k$ which are inverse limits of objects in $\mathcal{C}$ and with local homomorphisms.

If $F$ and $G$ are functors from $\mathcal{C}$ or $\mathcal{D}$ to the category of sets $\mathrm{Sets}$, we say that morphism of functors $F \to G$ is formally smooth if for every surjection $B \to A$ in $\mathcal{C}$ or $\mathcal{D}$, the induced map $F(B) \to F(A) \times_{G(A)}G(B)$ is surjective.

We can show that if $R\to S$ is a morphism in $\mathcal{C}$, then the morphism of functors $\mathrm{Hom}(S,-) \to \mathrm{Hom}(R,-)$ is formally smooth if and only if $S$ is a power series ring over $R$ (see for example M. Schlessinger, Functors of Artin rings, Proposition 2.5).

Is the same statement true if we work in $\mathcal{D}$ instead of $\mathcal{C}$, allowing power series with an infinite number of variables?