Choose a representable stack in groupoids $\mathcal{U}$ over $S$ and a $1$-morphism $f : \mathcal{U} \to \mathcal{H}_d(\mathcal{X})$ which is representable by algebraic spaces, smooth, and surjective. This is possible because $\mathcal{H}_d(\mathcal{X})$ is an algebraic stack by Lemma \ref{lemma-hilbert-stack-of-space}. Consider the $2$-fibre product $$ \mathcal{W} = \mathcal{H}_d(\mathcal{X}/\mathcal{Y}) \times_{\mathcal{H}_d(\mathcal{X}), f} \mathcal{U}. $$ Since $\mathcal{U}$ is representable (in particular a stack in setoids) it follows from Examples of Stacks, Lemma \ref{examples-stacks-lemma-faithful-hilbert} and Stacks, Lemma \ref{stacks-lemma-2-fibre-product-gives-stack-in-setoids} that $\mathcal{W}$ is a stack in setoids. The $1$-morphism $\mathcal{W} \to \mathcal{H}_d(\mathcal{X}/\mathcal{Y})$ is representable by algebraic spaces, smooth, and surjective as a base change of the morphism $f$ (see Algebraic Stacks, Lemmas \ref{algebraic-lemma-base-change-representable-by-spaces} and \ref{algebraic-lemma-base-change-representable-transformations-property}). Thus, if we can show that $\mathcal{W}$ is representable by an algebraic space, then the lemma follows from Algebraic Stacks, Lemma \ref{algebraic-lemma-smooth-surjective-morphism-implies-algebraic}. \medskip\noindent The diagonal of $\mathcal{Y}$ is representable by algebraic spaces according to Lemma \ref{lemma-flat-finite-presentation-surjective-diagonal}. We may apply Lemma \ref{lemma-relative-hilbert} to see that the $1$-morphism $$ \mathcal{H}_d(\mathcal{X}/\mathcal{Y}) \longrightarrow \mathcal{H}_d(\mathcal{X}) \times \mathcal{Y} $$ is representable by algebraic spaces. Consider the $2$-fibre product $$ \mathcal{V} = \mathcal{H}_d(\mathcal{X}/\mathcal{Y}) \times_{(\mathcal{H}_d(\mathcal{X}) \times \mathcal{Y}), f \times F} (\mathcal{U} \times \mathcal{X}). $$ The projection morphism $\mathcal{V} \to \mathcal{U} \times \mathcal{X}$ is representable by algebraic spaces as a base change of the last displayed morphism. Hence $\mathcal{V}$ is an algebraic space (see Bootstrap, Lemma \ref{bootstrap-lemma-representable-by-spaces-over-space} or Algebraic Stacks, Lemma \ref{algebraic-lemma-base-change-by-space-representable-by-space}). The $1$-morphism $\mathcal{V} \to \mathcal{U}$ fits into the following $2$-cartesian diagram $$ \xymatrix{ \mathcal{V} \ar[d] \ar[r] & \mathcal{X} \ar[d]^F \\ \mathcal{W} \ar[r] & \mathcal{Y} } $$ because $$ \mathcal{H}_d(\mathcal{X}/\mathcal{Y}) \times_{(\mathcal{H}_d(\mathcal{X}) \times \mathcal{Y}), f \times F} (\mathcal{U} \times \mathcal{X}) = (\mathcal{H}_d(\mathcal{X}/\mathcal{Y}) \times_{\mathcal{H}_d(\mathcal{X}), f} \mathcal{U}) \times_{\mathcal{Y}, F} \mathcal{X}. $$ Hence $\mathcal{V} \to \mathcal{W}$ is representable by algebraic spaces, surjective, flat, and locally of finite presentation as a base change of $F$. It follows that the same thing is true for the corresponding sheaves of sets associated to $\mathcal{V}$ and $\mathcal{W}$, see Algebraic Stacks, Lemma \ref{algebraic-lemma-map-fibred-setoids-property}. Thus we conclude that the sheaf associated to $\mathcal{W}$ is an algebraic space by Bootstrap, Theorem \ref{bootstrap-theorem-final-bootstrap}.