We will prove both statements at the same time. Namely, given an object $\xi = (U, U', a, i, x', \alpha)$ we will endow $\mathit{Aut}(\xi)$ with the structure of a quasi-coherent $\mathcal{O}_U$-module on $U_{spaces, \etale}$ and we will show that this structure is compatible with pullbacks. This will be sufficient by glueing of sheaves (Sites, Section \ref{sites-section-glueing-sheaves}) and the construction of $\mathcal{G}$ in the proof of Stacks, Lemma \ref{stacks-lemma-gerbe-abelian-auts} as the glueing of the automorphism sheaves $\mathit{Aut}(\xi)$ and the fact that it suffices to check a module is quasi-coherent after going to an \'etale covering (Properties of Spaces, Lemma \ref{spaces-properties-lemma-characterize-quasi-coherent}). \medskip\noindent We will describe the sheaf $\mathit{Aut}(\xi)$ using the same method as used in the proof of Lemma \ref{lemma-etale-local-lifts-isomorphic}. Consider the algebraic space $$ I' = U' \times_{x', \mathcal{X}', x'} U' $$ with projections $p' : I' \to U'$ and $q' : I' \to U'$. Over $I'$ there is a universal $2$-morphism $\gamma : x' \circ p' \to x' \circ q'$. The identity $x' \to x'$ defines a diagonal morphism $$ \xymatrix{ U' \ar[rr]_{\Delta'} & & I' \ar[ld]^{p'} \ar[rd]_{q'} \\ & U' & & U' } $$ such that the compositions $U' \to I' \to U'$ and $U' \to I' \to U'$ are the identity morphisms. We will denote the base change of $U', I', p', q', \Delta'$ to $\mathcal{X}$ by $U, I, p, q, \Delta$. Since $W' \to \mathcal{X}'$ is smooth, we see that $p' : I' \to U'$ is smooth as a base change. \medskip\noindent A section of $\mathit{Aut}(\xi)$ over $U$ is a morphism $\delta' : U' \to I'$ such that $\delta'|_U = \Delta$ and such that $p' \circ \delta' = \text{id}_{U'}$. To be explicit, $(\text{id}_U, q' \circ \delta', (\delta')^*\gamma) : \xi \to \xi$ is a formula for the corresponding automorphism. More generally, if $f : V \to U$ is an \'etale morphism, then there is a thickening $j : V \to V'$ and an \'etale morphism $f' : V' \to U'$ whose restriction to $V$ is $f$ and $f^*\xi$ corresponds to $(V, V', a \circ f, j, x' \circ f', f^*\alpha)$, see proof of Lemma \ref{lemma-gerbe-of-lifts-fibred}.  a section of $\mathit{Aut}(\xi)$ over $V$ is a morphism $\delta' : V' \to I'$ such that $\delta'|_V = \Delta \circ f$ and $p' \circ \delta' = f'$\footnote{A formula for the corresponding automorphism is $(\text{id}_V, h', (\delta')^*\gamma)$. Here $h' : V' \to V'$ is the unique (iso)morphism such that $h'|_V = \text{id}_V$ and such that $$ \xymatrix{ V' \ar[r]_{h'} \ar[rd]_{q' \circ \delta'} & V' \ar[d]^{f'} \\ & U' } $$ commutes. Uniqueness and existence of $h'$ by topological invariance of the \'etale site, see More on Morphisms of Spaces, Theorem \ref{spaces-more-morphisms-theorem-topological-invariance}. The reader may feel we should instead look at morphisms $\delta'' : V' \to V' \times_{\mathcal{X}'} V'$ with $\delta'' \circ j = \Delta_{V'/\mathcal{X}'}$ and $\text{pr}_1 \circ \delta'' = \text{id}_{V'}$. This would be fine too: as $V' \times_{\mathcal{X}'} V' \to I'$ is \'etale, the same topological invariance tells us that sending $\delta''$ to $\delta' = (V' \times_{\mathcal{X}'} V' \to I') \circ \delta''$ is a bijection between the two sets of morphisms.}. \medskip\noindent We conclude that $\mathit{Aut}(\xi)$ as a sheaf of sets agrees with the sheaf defined in More on Morphisms of Spaces, Remark \ref{spaces-more-morphisms-remark-another-special-case} for the thickenings $(U \subset U')$ and $(I \subset I')$ over $(U \subset U')$ via $\text{id}_{U'}$ and $p'$. The diagonal $\Delta'$ is a section of this sheaf and by acting on this section using More on Morphisms of Spaces, Lemma \ref{spaces-more-morphisms-lemma-action-sheaf} we get an isomorphism \begin{equation} \label{equation-isomorphism} \SheafHom_{\mathcal{O}_U}(\Delta^*\Omega_{I/U}, \mathcal{C}_{U/U'}) \longrightarrow \mathit{Aut}(\xi) \end{equation} on $U_{spaces, \etale}$. There three things left to check \begin{enumerate} \item the construction of (\ref{equation-isomorphism}) commutes with \'etale localization, \item $\SheafHom_{\mathcal{O}_U}(\Delta^*\Omega_{I/U}, \mathcal{C}_{U/U'})$ is a quasi-coherent module on $U$, \item the composition in $\mathit{Aut}(\xi)$ corresponds to addition of sections in this quasi-coherent module. \end{enumerate} We will check these in order. \medskip\noindent To see (1) we have to show that if $f : V \to U$ is \'etale, then (\ref{equation-isomorphism}) constructed using $\xi$ over $U$, restricts to the map (\ref{equation-isomorphism}) $$ \SheafHom_{\mathcal{O}_V}( \Delta_V^*\Omega_{V \times_\mathcal{X} V/V}, \mathcal{C}_{V/V'}) \to \mathit{Aut}(\xi|_V) $$ constructed using $\xi|_V$ over $V$ on $V_{spaces, \etale}$. This follows from the discussion in the footnote above and More on Morphisms of Spaces, Lemma \ref{spaces-more-morphisms-lemma-action-by-derivations-etale-localization}. \medskip\noindent Proof of (2). Since $p'$ is smooth, the morphism $I \to U$ is smooth, and hence the relative module of differentials $\Omega_{I/U}$ is finite locally free (More on Morphisms of Spaces, Lemma \ref{spaces-more-morphisms-lemma-smooth-omega-finite-locally-free}). On the other hand, $\mathcal{C}_{U/U'}$ is quasi-coherent (More on Morphisms of Spaces, Definition \ref{spaces-more-morphisms-definition-conormal-sheaf}). By Properties of Spaces, Lemma \ref{spaces-properties-lemma-properties-quasi-coherent} we conclude. \medskip\noindent Proof of (3). There exists a morphism $c' : I' \times_{p', U', q'} I' \to I'$ such that $(U', I', p', q', c')$ is a groupoid in algebraic spaces with identity $\Delta'$. See Algebraic Stacks, Lemma \ref{algebraic-lemma-map-space-into-stack} for example. Composition in $\mathit{Aut}(\xi)$ is induced by the morphism $c'$ as follows. Suppose we have two morphisms $$ \delta'_1, \delta'_2 : U' \longrightarrow I' $$ corresponding to sections of $\mathit{Aut}(\xi)$ over $U$ as above, in other words, we have $\delta'_i|U = \Delta_U$ and $p' \circ \delta'_i = \text{id}_{U'}$. Then the composition in $\mathit{Aut}(\xi)$ is $$ \delta'_1 \circ \delta'_2 = c'(\delta'_1 \circ q' \circ \delta'_2, \delta'_2) $$ We omit the detailed verification\footnote{The reader can see immediately that it is necessary to precompose $\delta'_1$ by $q' \circ \delta'_2$ to get a well defined $U'$-valued point of the fibre product $I' \times_{p', U', q'} I'$.}. Thus we are in the situation described in More on Groupoids in Spaces, Section \ref{spaces-more-groupoids-section-groupoid-sections} and the desired result follows from More on Groupoids in Spaces, Lemma \ref{spaces-more-groupoids-lemma-composition-is-addition}.