Note that by Groupoids in Spaces, Definition \ref{spaces-groupoids-definition-quotient-stack} the quotient stack $[X/G]$ is the stackification of the category fibred in groupoids associated to the ``presheaf in groupoids'' which associates to a scheme $T$ the groupoid $$ (X(T), G(T) \times X(T), s, t, c). $$ Since this ``presheaf in groupoids'' is isomorphic to $FA_d$ by Lemma \ref{lemma-represent-FAd} it suffices to prove that the $\mathcal{H}_d$ is the stackification of (the category fibred in groupoids associated to the ``presheaf in groupoids'') $FA_d$. To do this we first define a functor $$ \Spec : FA_d \longrightarrow \mathcal{H}_d $$ Recall that the fibre category of $\mathcal{H}_d$ over a scheme $T$ is the category of finite locally free morphisms $Z \to T$ of degree $d$. Thus given a scheme $T$ and a free $d$-dimensional $\Gamma(T, \mathcal{O}_T)$-algebra $m$ we may assign to this the object $$ Z = \underline{\Spec}_T(\mathcal{A}) $$ of $\mathcal{H}_{d, T}$ where $\mathcal{A} = \mathcal{O}_T^{\oplus d}$ endowed with a $\mathcal{O}_T$-algebra structure via $m$. Moreover, if $m'$ is a second such free $d$-dimensional $\Gamma(T, \mathcal{O}_T)$-algebra and if $\varphi : m \to m'$ is an isomorphism of these, then the induced $\mathcal{O}_T$-linear map $\varphi : \mathcal{O}_T^{\oplus d} \to \mathcal{O}_T^{\oplus d}$ induces an isomorphism $$ \varphi : \mathcal{A}' \longrightarrow \mathcal{A} $$ of quasi-coherent $\mathcal{O}_T$-algebras. Hence $$ \underline{\Spec}_T(\varphi) : \underline{\Spec}_T(\mathcal{A}) \longrightarrow \underline{\Spec}_T(\mathcal{A}') $$ is a morphism in the fibre category $\mathcal{H}_{d, T}$. We omit the verification that this construction is compatible with base change so we get indeed a functor $\Spec : FA_d \to \mathcal{H}_d$ as claimed above. \medskip\noindent To show that $\Spec : FA_d \to \mathcal{H}_d$ induces an equivalence between the stackification of $FA_d$ and $\mathcal{H}_d$ it suffices to check that \begin{enumerate} \item $\mathit{Isom}(m, m') = \mathit{Isom}(\Spec(m), \Spec(m'))$ for any $m, m' \in FA_d(T)$. \item for any scheme $T$ and any object $Z \to T$ of $\mathcal{H}_{d, T}$ there exists a covering $\{T_i \to T\}$ such that $Z|_{T_i}$ is isomorphic to $\Spec(m)$ for some $m \in FA_d(T_i)$, and \end{enumerate} see Stacks, Lemma \ref{stacks-lemma-stackify-groupoids}. The first statement follows from the observation that any isomorphism $$ \underline{\Spec}_T(\mathcal{A}) \longrightarrow \underline{\Spec}_T(\mathcal{A}') $$ is necessarily given by a global invertible matrix $g$ when $\mathcal{A} = \mathcal{A}' = \mathcal{O}_T^{\oplus d}$ as modules. To prove the second statement let $\pi : Z \to T$ be a finite locally free morphism of degree $d$. Then $\mathcal{A}$ is a locally free sheaf $\mathcal{O}_T$-modules of rank $d$. Consider the element $1 \in \Gamma(T, \mathcal{A})$. This element is nonzero in $\mathcal{A} \otimes_{\mathcal{O}_{T, t}} \kappa(t)$ for every $t \in T$ since the scheme $Z_t = \Spec(\mathcal{A} \otimes_{\mathcal{O}_{T, t}} \kappa(t))$ is nonempty being of degree $d > 0$ over $\kappa(t)$. Thus $1 : \mathcal{O}_T \to \mathcal{A}$ can locally be used as the first basis element (for example you can use Algebra, Lemma \ref{algebra-lemma-cokernel-flat} parts (1) and (2) to see this). Thus, after localizing on $T$ we may assume that there exists an isomorphism $\varphi : \mathcal{A} \to \mathcal{O}_T^{\oplus d}$ such that $1 \in \Gamma(\mathcal{A})$ corresponds to the first basis element. In this situation the multiplication map $\mathcal{A} \otimes_{\mathcal{O}_T} \mathcal{A} \to \mathcal{A}$ translates via $\varphi$ into a free $d$-dimensional algebra $m$ over $\Gamma(T, \mathcal{O}_T)$. This finishes the proof.