After replacing $X$, $S$, $T$, $Y$ by affine open subschemes we may assume the diagram in the statement of the lemma comes from applying $\Spec$ to a diagram $$ \vcenter{ \xymatrix{ A & B \\ R \ar[u] \ar[r] & C \ar[u] } } \quad\text{with primes}\quad \vcenter{ \xymatrix{ \mathfrak p_A & \mathfrak p_B \\ \mathfrak p_R \ar@{-}[u] \ar@{-}[r] & \mathfrak p_C \ar@{-}[u] } } $$ of Noetherian rings and finite type ring maps. In this proof every ring $E$ will be a Noetherian $R$-algebra endowed with a prime ideal $\mathfrak p_E$ lying over $\mathfrak p_R$ and all ring maps will be $R$-algebra maps compatible with the given primes. Moreover, if we write $E^\wedge$ we mean the completion of the localization of $E$ at $\mathfrak p_E$. We will also use without further mention that an \'etale ring map $E_1 \to E_2$ such that $\kappa(\mathfrak p_{E_1}) = \kappa(\mathfrak p_{E_2})$ induces an isomorphism $E_1^\wedge = E_2^\wedge$ by More on Algebra, Lemma \ref{more-algebra-lemma-flat-unramified}. \medskip\noindent With this notation $\sigma$ and $\varphi$ correspond to ring maps $$ \sigma : C \to R^\wedge \quad\text{and}\quad \varphi : A \longrightarrow (B \otimes_{C, \sigma} R^\wedge)^\wedge $$ Here is a picture $$ \xymatrix{ A \ar@/^1em/[rrr]^\varphi & B \ar[r] & B \otimes_{C, \sigma} R^\wedge \ar[r] & (B \otimes_{C, \sigma} R^\wedge)^\wedge \\ R \ar[r] \ar[u] & C \ar[r]^\sigma \ar[u] & R^\wedge \ar[u] \ar[ru] } $$ Observe that $R^\wedge$ is a G-ring by More on Algebra, Proposition \ref{more-algebra-proposition-Noetherian-complete-G-ring}. Thus $B \otimes_{C, \sigma} R^\wedge$ is a G-ring by More on Algebra, Proposition \ref{more-algebra-proposition-finite-type-over-G-ring}. By Lemma \ref{lemma-map-approximation} (translated into algebra) there exists an \'etale ring map $B \otimes_{C, \sigma} R^\wedge \to B'$ inducing an isomorphism $\kappa(\mathfrak p_{B \otimes_{C, \sigma} R^\wedge}) \to \kappa(\mathfrak p_{B'})$ and an $R$-algebra map $A \to B'$ such that the composition $$ A \to B' \to (B')^\wedge = (B \otimes_{C, \sigma} R^\wedge)^\wedge $$ is the same as $\varphi$ modulo $(\mathfrak p_{(B \otimes_{C, \sigma} R^\wedge)^\wedge})^N$. Thus we may replace $\varphi$ by this composition because the only way $\varphi$ enters the conclusion is via the commutativity requirement in part (5) of the statement of the lemma. Picture: $$ \xymatrix{ & & B' \ar[r] & (B')^\wedge \ar@{=}[d] \\ A \ar[rru] & B \ar[r] & B \otimes_{C, \sigma} R^\wedge \ar[r] \ar[u] & (B \otimes_{C, \sigma} R^\wedge)^\wedge \\ R \ar[r] \ar[u] & C \ar[r]^\sigma \ar[u] & R^\wedge \ar[u] \ar[ru] } $$ Next, we use that $R^\wedge$ is a filtered colimit of smooth $R$-algebras (Smoothing Ring Maps, Theorem \ref{smoothing-theorem-popescu}) because $R_{\mathfrak p_R}$ is a G-ring by assumption. Since $C$ is of finite presentation over $R$ we get a factorization $$ C \to R' \to R^\wedge $$ for some $R \to R'$ smooth, see Algebra, Lemma \ref{algebra-lemma-characterize-finite-presentation}. After increasing $R'$ we may assume there exists an \'etale $B \otimes_C R'$-algebra $B''$ whose base change to $B \otimes_{C, \sigma} R^\wedge$ is $B'$, see Algebra, Lemma \ref{algebra-lemma-etale}. Then $B'$ is the filtered colimit of these $B''$ and we conclude that after increasing $R'$ we may assume there is an $R$-algebra map $A \to B''$ such that $A \to B'' \to B'$ is the previously constructed map (same reference as above). Picture $$ \xymatrix{ & & B'' \ar[r] & B' \ar[r] & (B')^\wedge \ar@{=}[d] \\ A \ar[rru] & B \ar[r] & B \otimes_C R' \ar[r] \ar[u] & B \otimes_{C, \sigma} R^\wedge \ar[r] \ar[u] & (B \otimes_{C, \sigma} R^\wedge)^\wedge \\ R \ar[r] \ar[u] & C \ar[r] \ar[u] & R' \ar[r] \ar[u] & R^\wedge \ar[u] \ar[ru] } $$ and $$ B' = B'' \otimes_{(B \otimes_C R')} (B \otimes_{C, \sigma} R^\wedge) $$ This means that we may replace $C$ by $R'$, $\sigma : C \to R^\wedge$ by $R' \to R^\wedge$, and $B$ by $B''$ so that we simplify to the diagram $$ \xymatrix{ A \ar[r] & B \ar[r] & B \otimes_{C, \sigma} R^\wedge \\ R \ar[r] \ar[u] & C \ar[r]^\sigma \ar[u] & R^\wedge \ar[u] } $$ with $\varphi$ equal to the composition of the horizontal arrows followed by the canonical map from $B \otimes_{C, \sigma} R^\wedge$ to its completion. The final step in the proof is to apply Lemma \ref{lemma-map-approximation} (or its proof) one more time to $\Spec(C)$ and $\Spec(R)$ over $\Spec(R)$ and the map $C \to R^\wedge$. The lemma produces a ring map $C \to D$ such that $R \to D$ is \'etale, such that $\kappa(\mathfrak p_R) = \kappa(\mathfrak p_D)$, and such that $$ C \to D \to D^\wedge = R^\wedge $$ is equal to $\sigma : C \to R^\wedge$ modulo $(\mathfrak p_{R^\wedge})^N$. Then we can take $$ V = \Spec(D) \quad\text{and}\quad W = \Spec(B \otimes_C D) $$ as our solution to the problem posed by the lemma. Namely the diagram $$ \xymatrix{ A \ar[r] & B \otimes_{C, \sigma} R^\wedge \ar[r] & B \otimes_{C, \sigma} R^\wedge/(\mathfrak p_{R^\wedge})^N \ar@{=}[r] & B \otimes_C D/(\mathfrak p_D)^N \\ A \ar@{=}[u] \ar[r] & A \otimes_R D \ar[r] & B \otimes_R D \ar[r] & B \otimes_C D/(\mathfrak p_D)^N \ar@{=}[u] } $$ commutes because $C \to D \to D^\wedge = R^\wedge$ is equal to $\sigma$ modulo $(\mathfrak p_{R^\wedge})^N$. This proves part (5) and the other properties are immediate from the construction.