We prove $\Polarizedstack$ is a stack in groupoids over $\Spacesstack'_{fp, flat, proper}$ by checking conditions (1), (2), and (3) of Stacks, Definition \ref{stacks-definition-stack-in-groupoids}. We have already seen (1) in Lemma \ref{lemma-polarized-fibred-in-groupoids}. \medskip\noindent A covering of $\Spacesstack'_{fp, flat, proper}$ comes about in the following manner: Let $X \to S$ be an object of $\Spacesstack'_{fp, flat, proper}$. Suppose that $\{S_i \to S\}_{i \in I}$ is a covering of $\Sch_{fppf}$. Set $X_i = S_i \times_S X$. Then $\{(X_i \to S_i) \to (X \to S)\}_{i \in I}$ is a covering of $\Spacesstack'_{fp, flat, proper}$ and every covering of $\Spacesstack'_{fp, flat, proper}$ is isomorphic to one of these. Set $S_{ij} = S_i \times_S S_j$ and $X_{ij} = S_{ij} \times_S X$ so that $(X_{ij} \to S_{ij}) = (X_i \to S_i) \times_{(X \to S)} (X_j \to S_j)$. Next, suppose that $\mathcal{L}, \mathcal{N}$ are ample invertible sheaves on $X/S$ so that $(X \to S, \mathcal{L})$ and $(X \to S, \mathcal{N})$ are two objects of $\Polarizedstack$ over the object $(X \to S)$. To check descent for morphisms, we assume we have morphisms $(\text{id}, \text{id}, \varphi_i)$ from $(X_i \to S_i, \mathcal{L}|_{X_i})$ to $(X_i \to S_i, \mathcal{N}|_{X_i})$ whose base changes to morphisms from $(X_{ij} \to S_{ij}, \mathcal{L}|_{X_{ij}})$ to $(X_{ij} \to S_{ij}, \mathcal{N}|_{X_{ij}})$ agree. Then $\varphi_i : \mathcal{L}|_{X_i} \to \mathcal{N}|_{X_i}$ are isomorphisms of invertible modules over $X_i$ such that $\varphi_i$ and $\varphi_j$ restrict to the same isomorphisms over $X_{ij}$. By descent for quasi-coherent sheaves (Descent on Spaces, Proposition \ref{spaces-descent-proposition-fpqc-descent-quasi-coherent}) we obtain a unique isomorphism $\varphi : \mathcal{L} \to \mathcal{N}$ whose restriction to $X_i$ recovers $\varphi_i$. \medskip\noindent Decent for objects is proved in exactly the same manner. Namely, suppose that $\{(X_i \to S_i) \to (X \to S)\}_{i \in I}$ is a covering of $\Spacesstack'_{fp, flat, proper}$ as above. Suppose we have objects $(X_i \to S_i, \mathcal{L}_i)$ of $\Polarizedstack$ lying over $(X_i \to S_i)$ and a descent datum $$ (\text{id}, \text{id}, \varphi_{ij}) : (X_{ij} \to S_{ij}, \mathcal{L}_i|_{X_{ij}}) \to (X_{ij} \to S_{ij}, \mathcal{L}_j|_{X_{ij}}) $$ satisfying the obvious cocycle condition over $(X_{ijk} \to S_{ijk})$ for every triple of indices. Then by  descent for quasi-coherent sheaves (Descent on Spaces, Proposition \ref{spaces-descent-proposition-fpqc-descent-quasi-coherent}) we obtain a unique invertible $\mathcal{O}_X$-module $\mathcal{L}$ and isomorphisms $\mathcal{L}|_{X_i} \to \mathcal{L}_i$ recovering the descent datum $\varphi_{ij}$. To show that $(X \to S, \mathcal{L})$ is an object of $\Polarizedstack$ we have to prove that $\mathcal{L}$ is ample. This follows from Descent on Spaces, Lemma \ref{spaces-descent-lemma-descending-property-ample}. \medskip\noindent Since we already have seen that $\Spacesstack'_{fp, flat, proper}$ is a stack in groupoids over $\Sch_{fppf}$ (Lemma \ref{lemma-spaces-stack}) it now follows formally that $\Polarizedstack$ is a stack in groupoids over $\Sch_{fppf}$. See Stacks, Lemma \ref{stacks-lemma-stack-over-stack}.