By Lemma \ref{lemma-fully-faithful-cases} the functor (\ref{equation-descent-etale}) is fully faithful and the same remains the case after any base change $S \to S'$. Let $(V, \varphi)$ be a descent data relative to $X/S$ with $V \to X$ quasi-compact, separated, and \'etale. We can use Lemma \ref{lemma-reduce-to-affine} to see that it suffices to prove the effectivity  Zariski locally on $S$. In particular we may and do assume that $S$ is affine. \medskip\noindent If $S$ is affine we can find a directed set $\Lambda$ and an inverse system $X_\lambda \to S_\lambda$ of finite morphisms of affine schemes of finite type over $\Spec(\mathbf{Z})$ such that $(X \to S) = \lim (X_\lambda \to S_\lambda)$. See Algebra, Lemma \ref{algebra-lemma-limit-integral}. Since limits commute with limits we deduce that $X \times_S X = \lim X_\lambda \times_{S_\lambda} X_\lambda$ and  $X \times_S X \times_S X = \lim X_\lambda \times_{S_\lambda} X_\lambda \times_{S_\lambda} X_\lambda$. Observe that $V \to X$ is a morphism of finite presentation. Using Limits, Lemmas \ref{limits-lemma-descend-finite-presentation} we can find an $\lambda$ and a descent datum $(V_\lambda, \varphi_\lambda)$ relative to $X_\lambda/S_\lambda$ whose pullback to $X/S$ is $(V, \varphi)$. Of course it is enough to show that $(V_\lambda, \varphi_\lambda)$ is effective. Note that $V_\lambda$ is quasi-compact by construction. After possibly increasing $\lambda$ we may assume that $V_\lambda \to X_\lambda$ is separated and \'etale, see Limits, Lemma \ref{limits-lemma-descend-separated-finite-presentation} and \ref{limits-lemma-descend-etale}. Thus we may assume that $f$ is finite surjective and $S$ affine of finite type over $\mathbf{Z}$. \medskip\noindent Consider an open $S' \subset S$ such that the pullback $(V', \varphi')$ of $(V, \varphi)$ to $X' = X \times_S S'$ is effective. Below we will prove, that $S' \not = S$ implies there is a strictly larger open over which the descent datum is effective. Since $S$ is Noetherian (and hence has a Noetherian underlying topological space) this will finish the proof. Let $\xi \in S$ be a generic point of an irreducible component of the closed subset $Z = S \setminus S'$. If $\xi \in S'' \subset S$ is an open over which the descent datum is effective, then the descent datum is effective over $S' \cup S''$ by the glueing argument of the first paragraph. Thus in the rest of the proof we may replace $S$ by an affine open neighbourhood of $\xi$. \medskip\noindent After a first such replacement we may assume that $Z$ is irreducible with generic point $Z$. Let us endow $Z$ with the reduced induced closed subscheme structure. After another shrinking we may assume $X_Z = X \times_S Z = f^{-1}(Z) \to Z$ is flat, see Morphisms, Proposition \ref{morphisms-proposition-generic-flatness}. Let $(V_Z, \varphi_Z)$ be the pullback of the descent datum to $X_Z/Z$. By More on Morphisms, Lemma \ref{more-morphisms-lemma-separated-locally-quasi-finite-morphisms-fppf-descend} this descent datum is effective and we obtain an \'etale morphism $U_Z \to Z$ whose base change is isomorphic to $V_Z$ in a manner compatible with descent data. Of course $U_Z \to Z$ is quasi-compact and separated (Descent, Lemmas \ref{descent-lemma-descending-property-quasi-compact} and \ref{descent-lemma-descending-property-separated}). Thus after shrinking once more we may assume that $U_Z \to Z$ is finite, see Morphisms, Lemma \ref{morphisms-lemma-generically-finite}. \medskip\noindent Let $S = \Spec(A)$ and let $I \subset A$ be the prime ideal corresponding to $Z \subset S$. Let $(A^h, IA^h)$ be the henselization of the pair $(A, I)$. Denote $S^h = \Spec(A^h)$ and $Z^h = V(IA^h) \cong Z$. We claim that it suffices to show effectivity after base change to $S^h$. Namely, $\{S^h \to S, S' \to S\}$ is an fpqc covering ($A \to A^h$ is flat by More on Algebra, Lemma \ref{more-algebra-lemma-henselization-flat}) and by More on Morphisms, Lemma \ref{more-morphisms-lemma-separated-locally-quasi-finite-morphisms-fppf-descend} we have fpqc descent for separated \'etale morphisms. Namely, if $U^h \to S^h$ and $U' \to S'$ are the objects corresponding to the pullbacks $(V^h, \varphi^h)$ and $(V', \varphi')$, then the required isomorphisms $$ U^h \times_S S^h \to S^h \times_S V^h \quad\text{and}\quad U^h \times_S S' \to S^h \times_S U' $$ are obtained by the fully faithfulness pointed out in the first paragraph. In this way we reduce to the situation described in the next paragraph. \medskip\noindent Here $S = \Spec(A)$, $Z = V(I)$, $S' = S \setminus Z$ where $(A, I)$ is a henselian pair, we have $U' \to S'$ corresponding to the descent datum $(V', \varphi')$ and we have a finite \'etale morphism $U_Z \to Z$ corresponding to the descent datum $(V_Z, \varphi_Z)$. We no longer have that $A$ is of finite type over $\mathbf{Z}$; but the rest of the argument will not even use that $A$ is Noetherian. By More on Algebra, Lemma \ref{more-algebra-lemma-finite-etale-equivalence} we can find a finite \'etale morphism $U_{fin} \to S$ whose restriction to $Z$ is isomorphic to $U_Z \to Z$. Write $X = \Spec(B)$ and $Y = V(IB)$. Since $(B, IB)$ is a henselian pair (More on Algebra, Lemma \ref{more-algebra-lemma-integral-over-henselian-pair}) and since the restriction $V \to X$ to $Y$ is finite (as base change of $U_Z \to Z$) we see that there is a canonical disjoint union decomposition $$ V = V_{fin} \amalg V_{away} $$ were $V_{fin} \to X$ is finite and where $V_{away}$ has no points lying over $Y$. See Lemma \ref{lemma-split-henselian}. Using the uniqueness of this decomposition over $X \times_S X$ we see that $\varphi$ preserves it and we obtain $$ (V, \varphi) = (V_{fin}, \varphi_{fin}) \amalg (V_{away}, \varphi_{away}) $$ in the category of descent data. By More on Algebra, Lemma \ref{more-algebra-lemma-finite-etale-equivalence} there is a unique isomorphism $$ X \times_S U_{fin} \longrightarrow V_{fin} $$ compatible with the given isomorphism $Y \times_Z U_Z \to V \times_X Y$ over $Y$. By the uniqueness we see that this isomorphism is compatible with descent data, i.e., $(X \times_S U_{fin}, can) \cong (V_{fin}, \varphi_{fin})$. Denote $U'_{fin} = U_{fin} \times_S S'$. By fully faithfulness we obtain a morphism $U'_{fin} \to U'$ which is the inclusion of an open (and closed) subscheme. Then we set $U = U_{fin} \amalg_{U'_{fin}} U'$ (glueing of schemes as in Schemes, Section \ref{schemes-section-glueing-schemes}). The morphisms $X \times_S U_{fin} \to V$ and $X \times_S U' \to V$ glue to a morphism $X \times_S U \to V$ which is the desired isomorphism.