Pick $0 \in I$. Choose a surjective \'etale morphism $U_0 \to X_0$ where $U_0$ is an affine scheme (Properties of Spaces, Lemma \ref{spaces-properties-lemma-quasi-compact-affine-cover}). Set $U_i = X_i \times_{X_0} U_0$. Set $R_0 = U_0 \times_{X_0} U_0$ and $R_i = R_0 \times_{X_0} X_i$. Denote $s_i, t_i : R_i \to U_i$ and $s, t : R \to U$ the two projections. In the proof of Lemma \ref{lemma-directed-inverse-system-has-limit} we have seen that there exists a presentation $X = U/R$ with $U = \lim U_i$ and $R = \lim R_i$. Note that $U_i$ and $U$ are affine and that $R_i$ and $R$ are quasi-compact and separated (as $X_i$ is quasi-separated). Let $Y$ be an algebraic space over $S$ and let $Y \to X$ be a morphism of finite presentation. Set $V = U \times_X Y$. This is an algebraic space of finite presentation over $U$. Choose an affine scheme $W$ and a surjective \'etale morphism $W \to V$. Then $W \to Y$ is surjective \'etale as well. Set $R' = W \times_Y W$ so that $Y = W/R'$ (see Spaces, Section \ref{spaces-section-presentations}). Note that $W$ is a scheme of finite presentation over $U$ and that $R'$ is a scheme of finite presentation over $R$ (details omitted). By Limits, Lemma \ref{limits-lemma-descend-finite-presentation} we can find an index $i$ and a morphism of schemes $W_i \to U_i$ of finite presentation whose base change to $U$ gives $W \to U$. Similarly we can find, after possibly increasing $i$, a scheme $R'_i$ of finite presentation over $R_i$ whose base change to $R$ is $R'$. The projection morphisms $s', t' : R' \to W$ are morphisms over the projection morphisms $s, t : R \to U$. Hence we can view $s'$, resp.\ $t'$ as a morphism between schemes of finite presentation over $U$ (with structure morphism $R' \to U$ given by $R' \to R$ followed by $s$, resp.\ $t$). Hence we can apply Limits, Lemma \ref{limits-lemma-descend-finite-presentation} again to see that, after possibly increasing $i$, there exist morphisms $s'_i, t'_i : R'_i \to W_i$, whose base change to $U$ is $S', t'$. By Limits, Lemmas \ref{limits-lemma-descend-etale} and \ref{limits-lemma-descend-monomorphism} we may assume that $s'_i, t'_i$ are \'etale and that $j'_i : R'_i \to W_i \times_{X_i} W_i$ is a monomorphism (here we view $j'_i$ as a morphism of schemes of finite presentation over $U_i$ via one of the projections -- it doesn't matter which one). Setting $Y_i = W_i/R'_i$ (see Spaces, Theorem \ref{spaces-theorem-presentation}) we obtain an algebraic space of finite presentation over $X_i$ whose base change to $X$ is isomorphic to $Y$. \medskip\noindent This shows that every algebraic space of finite presentation over $X$ comes from an algebraic space of finite presentation over some $X_i$, i.e., it shows that the functor of the lemma is essentially surjective. To show that it is fully faithful, consider an index $0 \in I$ and two algebraic spaces $Y_0, Z_0$ of finite presentation over $X_0$. Set $Y_i = X_i \times_{X_0} Y_0$, $Y = X \times_{X_0} Y_0$, $Z_i = X_i \times_{X_0} Z_0$, and $Z = X \times_{X_0} Z_0$. Let $\alpha : Y \to Z$ be a morphism of algebraic spaces over $X$. Choose a surjective \'etale morphism $V_0 \to Y_0$ where $V_0$ is an affine scheme. Set $V_i = V_0 \times_{Y_0} Y_i$ and $V = V_0 \times_{Y_0} Y$ which are affine schemes endowed with surjective \'etale morphisms to $Y_i$ and $Y$. The composition $V \to Y \to Z \to Z_0$ comes from a (essentially unique) morphism $V_i \to Z_0$ for some $i \geq 0$ by Proposition \ref{proposition-characterize-locally-finite-presentation} (applied to $Z_0 \to X_0$ which is of finite presentation by assumption). After increasing $i$ the two compositions $$ V_i \times_{Y_i} V_i \to V_i \to Z_0 $$ are equal as this is true in the limit. Hence we obtain a (essentially unique) morphism $Y_i \to Z_0$. Since this is a morphism over $X_0$ it induces a morphism into $Z_i = Z_0 \times_{X_0} X_i$ as desired.