Assume (1). Let $T$ be a scheme and let $y \in Y(T)$. We have to show that $T \times_Y X$ is limit preserving over $T$ in the sense of Definition \ref{definition-locally-finite-presentation}. Hence we are reduced to proving that if $X$ is an algebraic space which is locally of finite presentation over $S$ as an algebraic space, then it is limit preserving as a functor $X : (\Sch/S)_{fppf}^{opp} \to \textit{Sets}$. To see this choose a presentation $X = U/R$, see Spaces, Definition \ref{spaces-definition-presentation}. It follows from Morphisms of Spaces, Definition \ref{spaces-morphisms-definition-locally-finite-presentation} that both $U$ and $R$ are schemes which are locally of finite presentation over $S$. Hence by Limits, Proposition \ref{limits-proposition-characterize-locally-finite-presentation} we have $$ U(T) = \colim U(T_i), \quad R(T) = \colim R(T_i) $$ whenever $T = \lim_i T_i$ in $(\Sch/S)_{fppf}$. It follows that the presheaf $$ (\Sch/S)_{fppf}^{opp} \longrightarrow \textit{Sets}, \quad W \longmapsto U(W)/R(W) $$ is limit preserving. Hence by Lemma \ref{lemma-sheafify-finite-presentation} its sheafification $X = U/R$ is limit preserving too. \medskip\noindent Assume (2). Choose a scheme $V$ and a surjective \'etale morphism $V \to Y$. Next, choose a scheme $U$ and a surjective \'etale morphism $U \to V \times_Y X$. By Lemma \ref{lemma-base-change-locally-finite-presentation} the transformation of functors $V \times_Y X \to V$ is limit preserving. By Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-etale-locally-finite-presentation} the morphism of algebraic spaces $U \to V \times_Y X$ is locally of finite presentation, hence limit preserving as a transformation of functors by the first part of the proof. By Lemma \ref{lemma-composition-locally-finite-presentation} the composition $U \to V \times_Y X \to V$ is limit preserving as a transformation of functors. Hence the morphism of schemes $U \to V$ is locally of finite presentation by Limits, Proposition \ref{limits-proposition-characterize-locally-finite-presentation} (modulo a set theoretic remark, see last paragraph of the proof). This means, by definition, that (1) holds. \medskip\noindent Set theoretic remark. Let $U \to V$ be a morphism of $(\Sch/S)_{fppf}$. In the statement of Limits, Proposition \ref{limits-proposition-characterize-locally-finite-presentation} we characterize $U \to V$ as being locally of finite presentation if for {\it all} directed inverse systems $(T_i, f_{ii'})$ of affine schemes over $V$ we have $U(T) = \colim V(T_i)$, but in the current setting we may only consider affine schemes $T_i$ over $V$ which are (isomorphic to) an object of $(\Sch/S)_{fppf}$. So we have to make sure that there are enough affines in $(\Sch/S)_{fppf}$ to make the proof work. Inspecting the proof of (2) $\Rightarrow$ (1) of Limits, Proposition \ref{limits-proposition-characterize-locally-finite-presentation} we see that the question reduces to the case that $U$ and $V$ are affine. Say $U = \Spec(A)$ and $V = \Spec(B)$. By construction of $(\Sch/S)_{fppf}$ the spectrum of any ring of cardinality $\leq |B|$ is isomorphic to an object of $(\Sch/S)_{fppf}$. Hence it suffices to observe that in the "only if" part of the proof of Algebra, Lemma \ref{algebra-lemma-characterize-finite-presentation} only $A$-algebras of cardinality $\leq |B|$ are used.