Recall that $\Polarizedstack$ is a limit preserving algebraic stack, see Quot, Lemma \ref{quot-lemma-polarized-limits}. By Limits of Stacks, Lemma \ref{stacks-limits-lemma-limit-preserving-diagonal} this implies that $\Delta : \Polarizedstack \to \Polarizedstack \times \Polarizedstack$ is limit preserving. Hence $\Delta$ is locally of finite presentation by Limits of Stacks, Proposition \ref{stacks-limits-proposition-characterize-locally-finite-presentation}. \medskip\noindent Let us prove that $\Delta$ is separated. To see this, it suffices to show that given an affine scheme $U$ and two objects $\upsilon = (Y, \mathcal{N})$ and $\chi = (X, \mathcal{L})$ of $\Polarizedstack$ over $U$, the algebraic space $$ \mathit{Isom}_{\Polarizedstack}(\upsilon, \chi) $$ is separated. The rule which to an isomorphism $\upsilon_T \to \chi_T$ assigns the underlying isomorphism $Y_T \to X_T$ defines a morphism $$ \mathit{Isom}_{\Polarizedstack}(\upsilon, \chi) \longrightarrow \mathit{Isom}_U(Y, X) $$ Since we have seen in Lemmas \ref{lemma-Mor-s-lfp} and \ref{lemma-Isom-in-Mor} that the target is a separated algebraic space, it suffices to prove that this morphism is separated. Given an isomorphism $f : Y_T \to X_T$ over some scheme $T/U$, then clearly $$ \mathit{Isom}_{\Polarizedstack}(\upsilon, \chi) \times_{\mathit{Isom}_U(Y, X), [f]} T = \mathit{Isom}(\mathcal{N}_T, f^*\mathcal{L}_T) $$ Here $[f] : T \to \mathit{Isom}_U(Y, X)$ indicates the $T$-valued point corresponding to $f$ and $\mathit{Isom}(\mathcal{N}_T, f^*\mathcal{L}_T)$ is the algebraic space discussed in Section \ref{section-hom-isom}. Since this algebraic space is affine over $U$, the claim implies $\Delta$ is separated. \medskip\noindent To finish the proof we show that $\Delta$ is quasi-compact. Since $\Delta$ is representable by algebraic spaces, it suffice to check the base change of $\Delta$ by a surjective smooth morphism $U \to \Polarizedstack \times \Polarizedstack$ is quasi-compact (see for example Properties of Stacks, Lemma \ref{stacks-properties-lemma-check-property-covering}). We can assume $U = \coprod U_i$ is a disjoint union of affine opens. Since $\Polarizedstack$ is limit preserving (see above), we see that $\Polarizedstack \to \Spec(\mathbf{Z})$ is locally of finite presentation, hence $U_i \to \Spec(\mathbf{Z})$ is locally of finite presentation (Limits of Stacks, Proposition \ref{stacks-limits-proposition-characterize-locally-finite-presentation} and Morphisms of Stacks, Lemmas \ref{stacks-morphisms-lemma-composition-finite-presentation} and \ref{stacks-morphisms-lemma-smooth-locally-finite-presentation}). In particular, $U_i$ is Noetherian affine. This reduces us to the case discussed in the next paragraph. \medskip\noindent In this paragraph, given a Noetherian affine scheme $U$ and two objects $\upsilon = (Y, \mathcal{N})$ and $\chi = (X, \mathcal{L})$ of $\Polarizedstack$ over $U$, we show the algebraic space $$ \mathit{Isom}_{\Polarizedstack}(\upsilon, \chi) $$ is quasi-compact. Since the connected components of $U$ are open and closed we may replace $U$ by these. Thus we may and do assume $U$ is connected. Let $u \in U$ be a point. Let $P$ be the Hilbert polynomial $n \mapsto \chi(Y_u, \mathcal{N}_u^{\otimes n})$, see Varieties, Lemma \ref{varieties-lemma-numerical-polynomial-from-euler}. Since $U$ is connected and since the functions $u \mapsto \chi(Y_u, \mathcal{N}_u^{\otimes n})$ are locally constant (see  Derived Categories of Schemes, Lemma \ref{perfect-lemma-chi-locally-constant-geometric}) we see that we get the same Hilbert polynomial in every point of $U$. Set $\mathcal{M} = \text{pr}_1^*\mathcal{N} \otimes_{\mathcal{O}_{Y \times_U X}} \text{pr}_2^*\mathcal{L}$ on $Y \times_U X$. Given $(f, \varphi) \in \mathit{Isom}_{\Polarizedstack}(\upsilon, \chi)(T)$ for some scheme $T$ over $U$ then for every $t \in T$ we have $$ \chi(Y_t, (\text{id} \times f)^*\mathcal{M}^{\otimes n}) = \chi(Y_t, \mathcal{N}_t^{\otimes n} \otimes_{\mathcal{O}_{Y_t}} f_t^*\mathcal{L}_t^{\otimes n}) = \chi(Y_t, \mathcal{N}_t^{\otimes 2n}) = P(2n) $$ where in the middle equality we use the isomorphism $\varphi : f^*\mathcal{L}_T \to \mathcal{N}_T$. Setting $P'(t) = P(2t)$ we find that the morphism $$ \mathit{Isom}_{\Polarizedstack}(\upsilon, \chi) \longrightarrow \mathit{Isom}_U(Y, X) $$ (see earlier) has image contained in the intersection $$ \mathit{Isom}_U(Y, X) \cap \mathit{Mor}^{P', \mathcal{M}}_U(Y, X) $$ The intersection is an intersection of open subspaces of $\mathit{Mor}_U(Y, X)$ (see Lemma \ref{lemma-Isom-in-Mor} and Remark \ref{remark-Mor-numerical}). Now $\mathit{Mor}^{P', \mathcal{M}}_U(Y, X)$ is a Noetherian algebraic space as it is of finite presentation over $U$ by Lemma \ref{lemma-Mor-qc-over-base}. Thus the intersection is a Noetherian algebraic space too. Since the morphism $$ \mathit{Isom}_{\Polarizedstack}(\upsilon, \chi) \longrightarrow \mathit{Isom}_U(Y, X) \cap \mathit{Mor}^{P', \mathcal{M}}_U(Y, X) $$ is affine (see above) we conclude.