The uniqueness we have seen in Lemma \ref{lemma-faithful}. Thus it suffices to prove existence. In this proof we will freely use the identifications of Equation (\ref{equation-localize-at-scheme-ringed}) as well as the result of Lemma \ref{lemma-relocalize-morphism-at-schemes}. \medskip\noindent Let $U \in \Ob(X_\etale)$, let $V \in \Ob(Y_\etale)$ and let $s \in g^{-1}h_V(U)$ be a section. We may think of $s$ as a map of sheaves $s : h_U \to g^{-1}h_V$. By Modules on Sites, Lemma \ref{sites-modules-lemma-relocalize-morphism-ringed-topoi} we obtain a commutative diagram of morphisms of ringed topoi $$ \xymatrix{ (\Sh(X_\etale/U), \mathcal{O}_U) \ar[rr]_-{(j, j^\sharp)} \ar[d]_{(g_s, g_s^\sharp)} & & (\Sh(X_\etale), \mathcal{O}_X) \ar[d]^{(g, g^\sharp)} \\ (\Sh(V_\etale), \mathcal{O}_V) \ar[rr] & & (\Sh(Y_\etale), \mathcal{O}_Y). } $$ By \'Etale Cohomology, Theorem \ref{etale-cohomology-theorem-fully-faithful} we obtain a unique morphism of schemes $f_s : U \to V$ such that $(g_s, g_s^\sharp)$ is $2$-isomorphic to $(f_{s, small}, f_s^\sharp)$. The construction $(U, V, s) \leadsto f_s$ just explained satisfies the following functoriality property: Suppose given morphisms $a : U' \to U$ in $X_\etale$ and $b : V' \to V$ in $Y_\etale$ and a map $s' : h_{U'} \to g^{-1}h_{V'}$ such that the diagram $$ \xymatrix{ h_{U'} \ar[d]_a \ar[r]_{s'} & g^{-1}h_{V'} \ar[d]^{g^{-1}b} \\ h_U \ar[r]^s & g^{-1}h_V } $$ commutes. Then the diagram $$ \xymatrix{ U' \ar[r]_-{f_{s'}} \ar[d]_a & u(V') \ar[d]^{u(b)} \\ U \ar[r]^-{f_s} & u(V) } $$ of schemes commutes. The reason this is true is that the same condition holds for the morphisms $(g_s, g_s^\sharp)$ constructed in Modules on Sites, Lemma \ref{sites-modules-lemma-relocalize-morphism-ringed-topoi} and the uniqueness in \'Etale Cohomology, Theorem \ref{etale-cohomology-theorem-fully-faithful}. \medskip\noindent The problem is to glue the morphisms $f_s$ to a morphism of algebraic spaces. To do this first choose a scheme $V$ and a surjective \'etale morphism $V \to Y$. This means that $h_V \to *$ is surjective and hence $g^{-1}h_V \to *$ is surjective too. This means there exists a scheme $U$ and a surjective \'etale morphism $U \to X$ and a morphism $s : h_U \to g^{-1}h_V$. Next, set $R = V \times_Y V$ and $R' = U \times_X U$. Then we get $g^{-1}h_R = g^{-1}h_V \times g^{-1}h_V$ as $g^{-1}$ is exact. Thus $s$ induces a morphism $s \times s : h_{R'} \to g^{-1}h_R$. Applying the constructions above we see that we get a commutative diagram of morphisms of schemes $$ \xymatrix{ R' \ar@<1ex>[d] \ar@<-1ex>[d] \ar[rr]_{f_{s \times s}} & & R \ar@<1ex>[d] \ar@<-1ex>[d] \\ U \ar[rr]^{f_s} & & V } $$ Since we have $X = U/R'$ and $Y = V/R$ (see Spaces, Lemma \ref{spaces-lemma-space-presentation}) we conclude that this diagram defines a morphism of algebraic spaces $f : X \to Y$ fitting into an obvious commutative diagram. Now we still have to show that $(f_{small}, f^\sharp)$ is $2$-isomorphic to $(g, g^\sharp)$. Let $t_V : f_{s, small}^{-1} \to g_s^{-1}$ and $t_R : f_{s \times s, small}^{-1} \to g_{s \times s}^{-1}$ be the $2$-isomorphisms which are given to us by the construction above. Let $\mathcal{G}$ be a sheaf on $Y_\etale$. Then we see that $t_V$ defines an isomorphism $$ f_{small}^{-1}\mathcal{G}|_{U_\etale} = f_{s, small}^{-1}\mathcal{G}|_{V_\etale} \xrightarrow{t_V} g_s^{-1}\mathcal{G}|_{V_\etale} = g^{-1}\mathcal{G}|_{U_\etale}. $$ Moreover, this isomorphism pulled back to $R'$ via either projection $R' \to U$ is the isomorphism $$ f_{small}^{-1}\mathcal{G}|_{R'_\etale} = f_{s \times s, small}^{-1}\mathcal{G}|_{R_\etale} \xrightarrow{t_R} g_{s \times s}^{-1}\mathcal{G}|_{R_\etale} = g^{-1}\mathcal{G}|_{R'_\etale}. $$ Since $\{U \to X\}$ is a covering in the site $X_{spaces, \etale}$ this means the first displayed isomorphism descends to an isomorphism $t : f_{small}^{-1}\mathcal{G} \to g^{-1}\mathcal{G}$ of sheaves (small detail omitted). The isomorphism is functorial in $\mathcal{G}$ since $t_V$ and $t_R$ are transformations of functors. Finally, $t$ is compatible with $f^\sharp$ and $g^\sharp$ as $t_V$ and $t_R$ are (some details omitted). This finishes the proof of the theorem.