The implication (1) $\Rightarrow$ (2) follows from Divisors on Spaces, Lemma \ref{spaces-divisors-lemma-ample-base-change}. Assume (2). To check $\mathcal{L}$ is ample on $X/Y$ we may work \'etale localy on $Y$, see Divisors on Spaces, Lemma \ref{spaces-divisors-lemma-relatively-ample-local}. Thus we may assume that $Y$ is a scheme and then we may in turn assume each $Y_i$ is a scheme too, see Topologies on Spaces, Lemma \ref{spaces-topologies-lemma-refine-fpqc-schemes}. In other words, we may assume that $\{Y_i \to Y\}$ is an fpqc covering of schemes. \medskip\noindent By Divisors on Spaces, Lemma \ref{spaces-divisors-lemma-relatively-ample-properties} we see that $X_i \to Y_i$ is representable (i.e., $X_i$ is a scheme), quasi-compact, and separated. Hence $f$ is quasi-compact and separated by Lemmas \ref{lemma-descending-property-quasi-compact} and \ref{lemma-descending-property-separated}. This means that $\mathcal{A} = \bigoplus_{d \geq 0} f_*\mathcal{L}^{\otimes d}$ is a quasi-coherent graded $\mathcal{O}_Y$-algebra (Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-pushforward}). Moreover, the formation of $\mathcal{A}$ commutes with flat base change by Cohomology of Spaces, Lemma \ref{spaces-cohomology-lemma-flat-base-change-cohomology}. In particular, if we set $\mathcal{A}_i = \bigoplus_{d \geq 0} f_{i, *}\mathcal{L}_i^{\otimes d}$ then we have $\mathcal{A}_i = g_i^*\mathcal{A}$. It follows that the natural maps $\psi_d : f^*\mathcal{A}_d \to \mathcal{L}^{\otimes d}$ of $\mathcal{O}_X$ pullback to give the natural maps $\psi_{i, d} : f_i^*(\mathcal{A}_i)_d \to \mathcal{L}_i^{\otimes d}$ of $\mathcal{O}_{X_i}$-modules. Since $\mathcal{L}_i$ is ample on $X_i/Y_i$ we see that for any point $x_i \in X_i$, there exists a $d \geq 1$ such that $f_i^*(\mathcal{A}_i)_d \to \mathcal{L}_i^{\otimes d}$ is surjective on stalks at $x_i$. This follows either directly from the definition of a relatively ample module or from Morphisms, Lemma \ref{morphisms-lemma-characterize-relatively-ample}. If $x \in |X|$, then we can choose an $i$ and an $x_i \in X_i$ mapping to $x$. Since $\mathcal{O}_{X, \overline{x}} \to \mathcal{O}_{X_i, \overline{x}_i}$ is flat hence faithfully flat, we conclude that for every $x \in |X|$ there exists a $d \geq 1$ such that $f^*\mathcal{A}_d \to \mathcal{L}^{\otimes d}$ is surjective on stalks at $x$. This implies that the open subset $U(\psi) \subset X$ of Divisors on Spaces, Lemma \ref{spaces-divisors-lemma-invertible-map-into-relative-proj} corresponding to the map $\psi : f^*\mathcal{A} \to \bigoplus_{d \geq 0} \mathcal{L}^{\otimes d}$ of graded $\mathcal{O}_X$-algebras is equal to $X$. Consider the corresponding morphism $$ r_{\mathcal{L}, \psi} : X \longrightarrow \underline{\text{Proj}}_Y(\mathcal{A}) $$ It is clear from the above that the base change of $r_{\mathcal{L}, \psi}$ to $Y_i$ is the morphism $r_{\mathcal{L}_i, \psi_i}$ which is an open immersion by Morphisms, Lemma \ref{morphisms-lemma-characterize-relatively-ample}. Hence $r_{\mathcal{L}, \psi}$ is an open immersion by Lemma \ref{lemma-descending-property-open-immersion}. Hence $X$ is a scheme and we conclude $\mathcal{L}$ is ample on $X/Y$ by Morphisms, Lemma \ref{morphisms-lemma-characterize-relatively-ample}.