Let $t : a_{small}^{-1} \to b_{small}^{-1}$ be the $2$-isomorphism. We may equivalently think of $t$ as a transformation $t : a_{spaces, \etale}^{-1} \to b_{spaces, \etale}^{-1}$ since there is not difference between sheaves on $X_\etale$ and sheaves on $X_{spaces, \etale}$. Choose a commutative diagram $$ \xymatrix{ U \ar[d]_p \ar[r]_\alpha  & V \ar[d]^q \\ X \ar[r]^a & Y } $$ where $U$ and $V$ are schemes, and $p$ and $q$ are surjective \'etale. Consider the diagram $$ \xymatrix{ h_U \ar[r]_-\alpha \ar@{=}[d] & a_{spaces, \etale}^{-1}h_V \ar[d]^t \\ h_U \ar@{..>}[r] & b_{spaces, \etale}^{-1}h_V } $$ Since the sheaf $b_{spaces, \etale}^{-1}h_V$ is isomorphic to $h_{V \times_{Y, b} X}$ we see that the dotted arrow comes from a morphism of schemes $\beta : U \to V$ fitting into a commutative diagram $$ \xymatrix{ U \ar[d]_p \ar[r]_\beta  & V \ar[d]^q \\ X \ar[r]^b & Y } $$ We claim that there exists a sequence of $2$-isomorphisms \begin{align*} (\alpha_{small}, \alpha^\sharp) & \cong (\alpha_{spaces, \etale}, \alpha^\sharp) \\ & \cong (a_{spaces, \etale, c}, a_c^\sharp) \\ & \cong (b_{spaces, \etale, d}, b_d^\sharp) \\ & \cong (\beta_{spaces, \etale}, \beta^\sharp) \\ & \cong (\beta_{small}, \beta^\sharp) \end{align*} The first and the last $2$-isomorphisms come from the identifications between sheaves on $U_{spaces, \etale}$ and sheaves on $U_\etale$ and similarly for $V$. The second and fourth $2$-isomorphisms are those of Lemma \ref{lemma-relocalize-morphism} with $c : U \to X \times_{a, Y} V$ induced by $\alpha$ and $d : U \to X \times_{b, Y} V$ induced by $\beta$. The middle $2$-isomorphism comes from the transformation $t$. Namely, the functor $a_{spaces, \etale, c}^{-1}$ corresponds to the functor $$ (\mathcal{H} \to h_V) \longmapsto (a_{spaces, \etale}^{-1}\mathcal{H} \times_{a_{spaces, \etale}^{-1}h_V, \alpha} h_U \to h_U) $$ and similarly for $b_{spaces, \etale, d}^{-1}$, see Sites, Lemma \ref{sites-lemma-relocalize-morphism}. This uses the identification of sheaves on $Y_{spaces, \etale}/V$ as arrows $(\mathcal{H} \to h_V)$ in $\Sh(Y_{spaces, \etale})$ and similarly for $U/X$, see Sites, Lemma \ref{sites-lemma-essential-image-j-shriek}. Via this identification the structure sheaf $\mathcal{O}_V$ corresponds to the pair $(\mathcal{O}_Y \times h_V \to h_V)$ and similarly for $\mathcal{O}_U$, see Modules on Sites, Lemma \ref{sites-modules-lemma-localize-compare}. Since $t$ switches $\alpha$ and $\beta$ we see that $t$ induces an isomorphism $$ t : a_{spaces, \etale}^{-1}\mathcal{H} \times_{a_{spaces, \etale}^{-1}h_V, \alpha} h_U \longrightarrow b_{spaces, \etale}^{-1}\mathcal{H} \times_{b_{spaces, \etale}^{-1}h_V, \beta} h_U $$ over $h_U$ functorially in $(\mathcal{H} \to h_V)$. Also, $t$ is compatible with $a_c^\sharp$ and $b_d^\sharp$ as $t$ is compatible with $a^\sharp$ and $b^\sharp$ by our description of the structure sheaves $\mathcal{O}_U$ and $\mathcal{O}_V$ above. Hence, the morphisms of ringed topoi $(\alpha_{small}, \alpha^\sharp)$ and $(\beta_{small}, \beta^\sharp)$ are $2$-isomorphic. By \'Etale Cohomology, Lemma \ref{etale-cohomology-lemma-faithful} we conclude $\alpha = \beta$! Since $p : U \to X$ is a surjection of sheaves it follows that $a = b$.