By Lemma \ref{lemma-flat-finite-presentation-surjective-diagonal} we see that the diagonal of $\mathcal{Y}$ is representable by algebraic spaces. Hence we only need to verify the existence of a $1$-morphism $f : \mathcal{V} \to \mathcal{Y}$ of stacks in groupoids over $(\Sch/S)_{fppf}$ with $\mathcal{V}$ representable and $f$ surjective and smooth. By Lemma \ref{lemma-hilbert-stack-relative-space} we know that $$ \coprod\nolimits_{d \geq 1} \mathcal{H}_d(\mathcal{X}/\mathcal{Y}) $$ is an algebraic stack. It follows from Lemma \ref{lemma-lci-locus-stack-in-groupoids} and Algebraic Stacks, Lemma \ref{algebraic-lemma-open-fibred-category-is-algebraic} that $$ \coprod\nolimits_{d \geq 1} \mathcal{H}_{d, lci}(\mathcal{X}/\mathcal{Y}) $$ is an algebraic stack as well. Choose a representable stack in groupoids $\mathcal{V}$ over $(\Sch/S)_{fppf}$ and a surjective and smooth $1$-morphism $$ \mathcal{V} \longrightarrow \coprod\nolimits_{d \geq 1} \mathcal{H}_{d, lci}(\mathcal{X}/\mathcal{Y}). $$ We claim that the composition $$ \mathcal{V} \longrightarrow \coprod\nolimits_{d \geq 1} \mathcal{H}_{d, lci}(\mathcal{X}/\mathcal{Y}) \longrightarrow \mathcal{Y} $$ is smooth and surjective which finishes the proof of the theorem. In fact, the smoothness will be a consequence of Lemmas \ref{lemma-limit-preserving} and \ref{lemma-lci-formally-smooth} and the surjectivity a consequence of Lemma \ref{lemma-lci-surjective}. We spell out the details in the following paragraph. \medskip\noindent By construction $\mathcal{V} \to \coprod\nolimits_{d \geq 1} \mathcal{H}_{d, lci}(\mathcal{X}/\mathcal{Y})$ is representable by algebraic spaces, surjective, and smooth (and hence also locally of finite presentation and formally smooth by the general principle Algebraic Stacks, Lemma \ref{algebraic-lemma-representable-transformations-property-implication} and More on Morphisms of Spaces, Lemma \ref{spaces-more-morphisms-lemma-smooth-formally-smooth}). Applying Lemmas \ref{lemma-representable-by-spaces-limit-preserving}, \ref{lemma-representable-by-spaces-formally-smooth}, and \ref{lemma-representable-by-spaces-surjective} we see that $\mathcal{V} \to \coprod\nolimits_{d \geq 1} \mathcal{H}_{d, lci}(\mathcal{X}/\mathcal{Y})$ is limit preserving on objects, formally smooth on objects, and surjective on objects. The $1$-morphism $\coprod\nolimits_{d \geq 1} \mathcal{H}_{d, lci}(\mathcal{X}/\mathcal{Y}) \to \mathcal{Y}$ is \begin{enumerate} \item limit preserving on objects: this is Lemma \ref{lemma-limit-preserving} for $\mathcal{H}_d(\mathcal{X}/\mathcal{Y}) \to \mathcal{Y}$ and we combine it with Lemmas \ref{lemma-lci-locus-stack-in-groupoids}, \ref{lemma-open-immersion-limit-preserving}, and \ref{lemma-composition-limit-preserving} to get it for $\mathcal{H}_{d, lci}(\mathcal{X}/\mathcal{Y}) \to \mathcal{Y}$, \item formally smooth on objects by Lemma \ref{lemma-lci-formally-smooth}, and \item surjective on objects by Lemma \ref{lemma-lci-surjective}. \end{enumerate} Using Lemmas \ref{lemma-composition-limit-preserving}, \ref{lemma-composition-formally-smooth}, and \ref{lemma-composition-surjective} we conclude that the composition $\mathcal{V} \to \mathcal{Y}$ is limit preserving on objects, formally smooth on objects, and surjective on objects. Using Lemmas \ref{lemma-representable-by-spaces-limit-preserving}, \ref{lemma-representable-by-spaces-formally-smooth}, and \ref{lemma-representable-by-spaces-surjective} we see that $\mathcal{V} \to \mathcal{Y}$ is locally of finite presentation, formally smooth, and surjective. Finally, using (via the general principle Algebraic Stacks, Lemma \ref{algebraic-lemma-representable-transformations-property-implication}) the infinitesimal lifting criterion (More on Morphisms of Spaces, Lemma \ref{spaces-more-morphisms-lemma-smooth-formally-smooth}) we see that $\mathcal{V} \to \mathcal{Y}$ is smooth and we win.