Case I: $\mathcal{X}$ is representable (by a scheme). Say $\mathcal{X} = (\Sch/X)_{fppf}$ for some scheme $X$ over $S$. Unwinding the definitions we have to prove the following: Given a Noetherian complete local $S$-algebra $R$ with $R/\mathfrak m$ of finite type over $S$ we have $$ \Mor_S(\Spec(R), X) \longrightarrow \lim \Mor_S(\Spec(R/\mathfrak m^n), X) $$ is bijective. This follows from Formal Spaces, Lemma \ref{formal-spaces-lemma-map-into-scheme}. \medskip\noindent Case II. $\mathcal{X}$ is representable by an algebraic space. Say $\mathcal{X}$ is representable by $X$. Again we have to show that $$ \Mor_S(\Spec(R), X) \longrightarrow \lim \Mor_S(\Spec(R/\mathfrak m^n), X) $$ is bijective for $R$ as above. This is Formal Spaces, Lemma \ref{formal-spaces-lemma-map-into-algebraic-space}. \medskip\noindent Case III: General case of an algebraic stack. A general remark is that the left and right hand side of (\ref{equation-approximation}) are categories fibred in groupoids over the category of affine schemes over $S$ which are spectra of Noetherian complete local rings with residue field of finite type over $S$. We will also see in the proof below that they form stacks for a certain topology on this category. \medskip\noindent We first prove fully faithfulness. Let $R$ be a Noetherian complete local $S$-algebra with $k = R/\mathfrak m$ of finite type over $S$. Let $x, x'$ be objects of $\mathcal{X}$ over $R$. As $\mathcal{X}$ is an algebraic stack $\mathit{Isom}(x, x')$ is representable by an algebraic space $I$ over $\Spec(R)$, see Algebraic Stacks, Lemma \ref{algebraic-lemma-representable-diagonal}. Applying Case II to $I$ over $\Spec(R)$ implies immediately that (\ref{equation-approximation}) is fully faithful on fibre categories over $\Spec(R)$. Hence the functor is fully faithful by Categories, Lemma \ref{categories-lemma-equivalence-fibred-categories}. \medskip\noindent Essential surjectivity. Let $\xi = (R, \xi_n, f_n)$ be a formal object of $\mathcal{X}$. Choose a scheme $U$ over $S$ and a surjective smooth morphism $f : (\Sch/U)_{fppf} \to \mathcal{X}$. For every $n$ consider the fibre product $$ (\Sch/\Spec(R/\mathfrak m^n))_{fppf} \times_{\xi_n, \mathcal{X}, f} (\Sch/U)_{fppf} $$ By assumption this is representable by an algebraic space $V_n$ surjective and smooth over $\Spec(R/\mathfrak m^n)$. The morphisms $f_n : \xi_n \to \xi_{n + 1}$ induce cartesian squares $$ \xymatrix{ V_{n + 1} \ar[d] & V_n \ar[d] \ar[l] \\ \Spec(R/\mathfrak m^{n + 1}) & \Spec(R/\mathfrak m^n) \ar[l] } $$ of algebraic spaces. By Spaces over Fields, Lemma \ref{spaces-over-fields-lemma-smooth-separable-closed-points-dense} we can find a finite separable extension $k \subset k'$ and a point $v'_1 : \Spec(k') \to V_1$ over $k$. Let $R \subset R'$ be the finite \'etale extension whose residue field extension is $k \subset k'$ (exists and is unique by Algebra, Lemmas \ref{algebra-lemma-henselian-cat-finite-etale} and \ref{algebra-lemma-complete-henselian}). By the infinitesimal lifting criterion of smoothness (see More on Morphisms of Spaces, Lemma \ref{spaces-more-morphisms-lemma-smooth-formally-smooth}) applied to $V_n \to \Spec(R/\mathfrak m^n)$ for $n = 2, 3, 4, \ldots$ we can successively find morphisms $v'_n : \Spec(R'/(\mathfrak m')^n) \to V_n$ over $\Spec(R/\mathfrak m^n)$ fitting into commutative diagrams $$ \xymatrix{ \Spec(R'/(\mathfrak m')^{n + 1}) \ar[d]_{v'_{n + 1}} & \Spec(R'/(\mathfrak m')^n) \ar[d]^{v'_n} \ar[l] \\ V_{n + 1} & V_n \ar[l] } $$ Composing with the projection morphisms $V_n \to U$ we obtain a compatible system of morphisms $u'_n : \Spec(R'/(\mathfrak m')^n) \to U$. By Case I the family $(u'_n)$ comes from a unique morphism $u' : \Spec(R') \to U$. Denote $x'$ the object of $\mathcal{X}$ over $\Spec(R')$ we get by applying the $1$-morphism $f$ to $u'$. By construction, there exists a morphism of formal objects $$ (\ref{equation-approximation})(x') = (R', x'|_{\Spec(R'/(\mathfrak m')^n)}, \ldots) \longrightarrow (R, \xi_n, f_n) $$ lying over $\Spec(R') \to \Spec(R)$. Note that $R' \otimes_R R'$ is a finite product of spectra of Noetherian complete local rings to which our current discussion applies. Denote $p_0, p_1 : \Spec(R' \otimes_R R') \to \Spec(R')$ the two projections. By the fully faithfulness shown above there exists a canonical isomorphism $\varphi : p_0^*x' \to p_1^*x'$ because we have such isomorphisms over $\Spec((R' \otimes_R R')/\mathfrak m^n(R' \otimes_R R'))$. We omit the proof that the isomorphism $\varphi$ satisfies the cocycle condition (see Stacks, Definition \ref{stacks-definition-descent-data}). Since $\{\Spec(R') \to \Spec(R)\}$ is an fppf covering we conclude that $x'$ descends to an object $x$ of $\mathcal{X}$ over $\Spec(R)$. We omit the proof that $x_n$ is the restriction of $x$ to $\Spec(R/\mathfrak m^n)$.