In this proof unadorned products are over $\Spec(k)$. By Proposition \ref{proposition-hilb-d} the scheme $H = \underline{\Hilbfunctor}^g_{X/k}$ exists. Consider the universal divisor $D_{univ} \subset H \times X$ and the associated invertible sheaf $\mathcal{O}(D_{univ})$, see Remark \ref{remark-universal-object-hilb-d}. We adjust by tensoring with the pullback via $\sigma_H : H \to H \times X$ to get $$ \mathcal{L}_H = \mathcal{O}(D_{univ}) \otimes_{\mathcal{O}_{H \times X}} \text{pr}_H^*\sigma_H^*\mathcal{O}(D_{univ})^{\otimes -1} \in \Picardfunctor_{X/k, \sigma}(H) $$ By the Yoneda lemma (Categories, Lemma \ref{categories-lemma-yoneda}) the invertible sheaf $\mathcal{L}_H$ defines a natural transformation $$ h_H \longrightarrow \Picardfunctor_{X/k, \sigma} $$ Because $F$ is an open subfuctor, there exists a maximal open $W \subset H$ such that $\mathcal{L}_H|_{W \times X}$ is in $F(W)$. Of course, this open is nothing else than the open subscheme constructed in Derived Categories of Schemes, Lemma \ref{perfect-lemma-open-where-cohomology-in-degree-i-rank-r-geometric} with $i = 0$ and $r = 1$ for the morphism $H \times X \to H$ and the sheaf $\mathcal{F} = \mathcal{O}(D_{univ})$. Applying the Yoneda lemma again we obtain a commutative diagram $$ \xymatrix{ h_W \ar[d] \ar[r] & F \ar[d] \\ h_H \ar[r] & \Picardfunctor_{X/k, \sigma} } $$ To finish the proof we will show that the top horizontal arrow is an isomorphism. \medskip\noindent Let $\mathcal{L} \in F(T) \subset \Picardfunctor_{X/k, \sigma}(T)$. Let $\mathcal{N}$ be the invertible $\mathcal{O}_T$-module such that $Rf_{T, *}\mathcal{L} \cong \mathcal{N}[0]$. The adjunction map $$ f_T^*\mathcal{N} \longrightarrow \mathcal{L} \quad\text{corresponds to a section }s\text{ of}\quad \mathcal{L} \otimes f_T^*\mathcal{N}^{\otimes -1} $$ on $X_T$. Claim: The zero scheme of $s$ is a relative effective Cartier divisor $D$ on $(T \times X)/T$ finite locally free of degree $g$ over $T$. \medskip\noindent Let us finish the proof of the lemma admitting the claim. Namely, $D$ defines a morphism $m : T \to H$ such that $D$ is the pullback of $D_{univ}$. Then $$ (m \times \text{id}_X)^*\mathcal{O}(D_{univ}) \cong \mathcal{O}_{T \times X}(D) $$ Hence $(m \times \text{id}_X)^*\mathcal{L}_H$ and $\mathcal{O}(D)$ differ by the pullback of an invertible sheaf on $H$. This in particular shows that $m : T \to H$ factors through the open $W \subset H$ above. Moreover, it follows that these invertible modules define, after adjusting by pullback via $\sigma_T$ as above, the same element of $\Picardfunctor_{X/k, \sigma}(T)$. Chasing diagrams using Yoneda's lemma we see that $m \in h_W(T)$ maps to $\mathcal{L} \in F(T)$. We omit the verification that the rule $F(T) \to h_W(T)$, $\mathcal{L} \mapsto m$ defines an inverse of the transformation of functors above. \medskip\noindent Proof of the claim. Since $D$ is a locally principal closed subscheme of $T \times X$, it suffices to show that the fibres of $D$ over $T$ are effective Cartier divisors, see Lemma \ref{lemma-divisors-on-curves} and Divisors, Lemma \ref{divisors-lemma-fibre-Cartier}. Because taking cohomology of $\mathcal{L}$ commutes with base change (Derived Categories of Schemes, Lemma \ref{perfect-lemma-flat-proper-perfect-direct-image-general}) we reduce to $T = \Spec(K)$ where $K/k$ is a field extension. Then $\mathcal{L}$ is an invertible sheaf on $X_K$ with $H^0(X_K, \mathcal{L}) = K$ and $H^1(X_K, \mathcal{L}) = 0$. Thus $$ \deg(\mathcal{L}) = \chi(X_K, \mathcal{L}) - \chi(X_K, \mathcal{O}_{X_K}) = 1 - (1 - g) = g $$ See Varieties, Definition \ref{varieties-definition-degree-invertible-sheaf}. To finish the proof we have to show a nonzero section of $\mathcal{L}$ defines an effective Cartier divisor on $X_K$. This is clear.