A formal object $\xi = (R, \xi_n)$ of $F$ consists of a Noetherian complete local $S$-algebra $R$ whose residue field is of finite type over $S$, together with elements $\xi_n \in F(\Spec(R/\mathfrak m^n))$ for all $n$ such that $\xi_{n + 1}|_{\Spec(R/\mathfrak m^n)} = \xi_n$. By the discussion in the proof of Lemma \ref{lemma-rs} we see that either $\xi$ is a formal object of $U'$ or a formal object of $W$. In the first case we see that $\xi$ is effective by Lemma \ref{lemma-effective}. The second case is the interesting case. Set $V = \Spec(R)$. We will construct an element $(Z, u', \hat x) \in F(V)$ whose image in $F(\Spec(R/\mathfrak m^n))$ is $\xi_n$ for all $n \geq 1$. \medskip\noindent We may view the collection of elements $\xi_n$ as a morphism $$ \xi : \text{Spf}(R) \longrightarrow W $$ of locally Noetherian formal algebraic spaces over $S$. Observe that $\xi$ is {\it not} an adic morphism in general. To fix this, let $I \subset R$ be the ideal corresponding to the formal closed subspace $$ \text{Spf}(R) \times_{\xi, W} W_{red} \subset \text{Spf}(R) $$ Note that $I \subset \mathfrak m_R$. Set $Z = V(I) \subset V = \Spec(R)$. Since $R$ is $\mathfrak m_R$-adically complete it is a fortiori $I$-adically complete (Algebra, Lemma \ref{algebra-lemma-complete-by-sub}). Moreover, we claim that for each $n \geq 1$ the morphism $$ \xi|_{\text{Spf}(R/I^n)} : \text{Spf}(R/I^n) \longrightarrow W $$ actually comes from a morphism $$ \xi'_n :  \Spec(R/I^n) \longrightarrow W $$ Namely, this follows from writing $W = \colim W_n$ as in the proof of Lemma \ref{lemma-rs}, noticing that $\xi|_{\text{Spf}(R/I^n)}$ maps into $W_n$, and applying Formal Spaces, Lemma \ref{formal-spaces-lemma-map-into-algebraic-space} to algebraize this to a morphism $\Spec(R/I^n) \to W_n$ as desired. Let us denote $\text{Spf}'(R) = V_{/Z}$ the formal spectrum of $R$ endowed with the $I$-adic topology -- equivalently the formal completion of $V$ along $Z$. Using the morphisms $\xi'_n$ we obtain an adic morphism $$ \hat x = (\xi'_n) : \text{Spf}'(R) \longrightarrow W $$ of locally Noetherian formal algebraic spaces over $S$. Consider the base change $$ \text{Spf}'(R) \times_{\hat x, W, g} X'_{/T'} \longrightarrow \text{Spf}'(R) $$ This is a formal modification by Algebraization of Formal Spaces, Lemma \ref{restricted-lemma-base-change-formal-modification}. Hence by the main theorem on dilatations (Algebraization of Formal Spaces, Theorem \ref{restricted-theorem-dilatations}) we obtain a proper morphism $$ V' \longrightarrow V = \Spec(R) $$ which is an isomorphism over $\Spec(R) \setminus V(I)$ and whose completion recovers the formal modification above, in other words $$ V' \times_{\Spec(R)} \Spec(R/I^n) = \Spec(R/I^n) \times_{\xi'_n, W, g} X'_{/T'} $$ This in particular tells us we have a compatible system of morphisms $$ V' \times_{\Spec(R)} \Spec(R/I^n) \longrightarrow X' \times_S \Spec(R/I^n) $$ Hence by Grothendieck's algebraization theorem (in the form of More on Morphisms of Spaces, Lemma \ref{spaces-more-morphisms-lemma-algebraize-morphism}) we obtain a morphism $$ x' : V' \to X' $$ over $S$ recovering the morphisms displayed above. Then finally setting $u' : V \setminus Z \to X'$ the restriction of $x'$ to $V \setminus Z \subset V'$ gives the third component of our desired element $(Z, u', \hat x) \in F(V)$.