The proof given here is an example of applying algebraization and approximation. We proceed in a number of stages. \medskip\noindent Essential surjectivity when $A$ is a complete local Noetherian ring. Let $X_n = X \times_{\Spec(A)} \Spec(A/\mathfrak m^{n + 1})$. By \'Etale Morphisms, Theorem \ref{etale-theorem-remarkable-equivalence} the inclusions $$ X_0 \to X_1 \to X_2 \to \ldots $$ induce equivalence of categories between the category of schemes \'etale over $X_0$ and the category of schemes \'etale over $X_n$. Moreover, if $U_n \to X_n$ corresponds to a finite \'etale morphism $U_0 \to X_0$, then $U_n \to X_n$ is finite too, for example by More on Morphisms, Lemma \ref{more-morphisms-lemma-thicken-property-morphisms-cartesian}. In this case the morphism $U_0 \to \Spec(A/\mathfrak m)$ is proper as $X_0$ is proper over $A/\mathfrak m$. Thus we may apply Grothendieck's algebraization theorem (in the form of Cohomology of Schemes, Lemma \ref{coherent-lemma-algebraize-formal-scheme-finite-over-proper}) to see that there is a finite morphism $U \to X$ whose restriction to $X_0$ recovers $U_0$. By More on Morphisms, Lemma \ref{more-morphisms-lemma-check-smoothness-on-infinitesimal-nbhds} we see that $U \to X$ is \'etale at every point of $U_0$. However, since every point of $U$ specializes to a point of $U_0$ (as $U$ is proper over $A$), we conclude that $U \to X$ is \'etale. In this way we conclude the functor is essentially surjective. \medskip\noindent Fully faithfulness when $A$ is a complete local Noetherian ring. Let $U \to X$ and $V \to X$ be finite \'etale morphisms and let $\varphi_0 : U_0 \to V_0$ be a morphism over $X_0$. Look at the morphism $$ \Gamma_{\varphi_0} : U_0 \longrightarrow U_0 \times_{X_0} V_0 $$ This morphism is both finite \'etale and a closed immersion. By essential surjectivity applied to $X = U \times_X V$ we find a finite \'etale morphism $W \to U \times_X V$ whose special fibre is isomorphic to $\Gamma_{\varphi_0}$. Consider the projection $W \to U$. It is finite \'etale and an isomorphism over $U_0$ by construction. By \'Etale Morphisms, Lemma \ref{etale-lemma-finite-etale-one-point} $W \to U$ is an isomorphism in an open neighbourhood of $U_0$. Thus it is an isomorphism and the composition $\varphi : U \cong W \to V$ is the desired lift of $\varphi_0$. \medskip\noindent Essential surjectivity when $A$ is a henselian local Noetherian G-ring. Let $U_0 \to X_0$ be a finite \'etale morphism. Let $A^\wedge$ be the completion of $A$ with respect to the maximal ideal. Let $X^\wedge$ be the base change of $X$ to $A^\wedge$. By the result above there exists a finite \'etale morphism $V \to X^\wedge$ whose special fibre is $U_0$. Write $A^\wedge = \colim A_i$ with $A \to A_i$ of finite type. By Limits, Lemma \ref{limits-lemma-descend-finite-presentation} there exists an $i$ and a finitely presented morphism $U_i \to X_{A_i}$ whose base change to $X^\wedge$ is $V$. After increasing $i$ we may assume that $U_i \to X_{A_i}$ is finite and \'etale (Limits, Lemmas \ref{limits-lemma-descend-finite-finite-presentation} and \ref{limits-lemma-descend-etale}). Writing $$ A_i = A[x_1, \ldots, x_n]/(f_1, \ldots, f_m) $$ the ring map $A_i \to A^\wedge$ can be reinterpreted as a solution $(a_1, \ldots, a_n)$ in $A^\wedge$ for the system of equations $f_j = 0$. By Smoothing Ring Maps, Theorem \ref{smoothing-theorem-approximation-property} we can approximate this solution (to order $11$ for example) by a solution $(b_1, \ldots, b_n)$ in $A$. Translating back we find an $A$-algebra map $A_i \to A$ which gives the same closed point as the original map $A_i \to A^\wedge$ (as $11 > 1$). The base change $U \to X$ of $V \to X_{A_i}$ by this ring map will therefore be a finite \'etale morphism whose special fibre is isomorphic to $U_0$. \medskip\noindent Fully faithfulness when $A$ is a henselian local Noetherian G-ring. This can be deduced from essential surjectivity in exactly the same manner as was done in the case that $A$ is complete Noetherian. \medskip\noindent General case. Let $(A, \mathfrak m)$ be a henselian local ring. Set $S = \Spec(A)$ and denote $s \in S$ the closed point. By Limits, Lemma \ref{limits-lemma-proper-limit-of-proper-finite-presentation-noetherian} we can write $X \to \Spec(A)$ as a cofiltered limit of proper morphisms $X_i \to S_i$ with $S_i$ of finite type over $\mathbf{Z}$. For each $i$ let $s_i \in S_i$ be the image of $s$. Since $S = \lim S_i$ and $A = \mathcal{O}_{S, s}$ we have $A = \colim \mathcal{O}_{S_i, s_i}$. The ring $A_i = \mathcal{O}_{S_i, s_i}$ is a Noetherian local G-ring (More on Algebra, Proposition \ref{more-algebra-proposition-ubiquity-G-ring}). By More on Algebra, Lemma \ref{more-algebra-lemma-henselization-colimit} we see that $A = \colim A_i^h$. By More on Algebra, Lemma \ref{more-algebra-lemma-henselization-G-ring} the rings $A_i^h$ are G-rings. Thus we see that $A = \colim A_i^h$ and $$ X = \lim (X_i \times_{S_i} \Spec(A_i^h)) $$ as schemes. The category of schemes finite \'etale over $X$ is the limit of the category of schemes finite \'etale over $X_i \times_{S_i} \Spec(A_i^h)$ (by Limits, Lemmas \ref{limits-lemma-descend-finite-presentation}, \ref{limits-lemma-descend-finite-finite-presentation}, and \ref{limits-lemma-descend-etale}) The same thing is true for schemes finite \'etale over $X_0 = \lim (X_i \times_{S_i} s_i)$. Thus we formally deduce the result for $X / \Spec(A)$ from the result for the $(X_i \times_{S_i} \Spec(A_i^h)) / \Spec(A_i^h)$ which we dealt with above.