Proof.
Consider the category \mathcal{C} whose objects are the schemes U over S such that every quasi-compact open of U has finitely many irreducible components and whose morphisms are those morphisms g : U \to V of schemes over S such that every generic point of an irreducible component of U is mapped to the generic point of an irreducible component of V. We have already shown that
for U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C}) we have a normalization morphism \nu _ U : U^\nu \to U as in Morphisms, Definition 29.54.1,
for U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C}) the morphism \nu _ U is integral and U^\nu is a normal scheme, see Morphisms, Lemma 29.54.5,
for every g : U \to V \in \text{Arrows}(\mathcal{C}) there is a unique morphism g^\nu : U^\nu \to V^\nu such that \nu _ V \circ g^\nu = g \circ \nu _ U, see Morphisms, Lemma 29.54.5 part (4) applied to the composition X^\nu \to X \to Y,
if V \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C}) and g : U \to V is étale or smooth, then U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C}) and g \in \text{Arrows}(\mathcal{C}) and the morphism g^\nu induces an isomorphism U^\nu \to U \times _ V V^\nu , see Lemma 67.49.4 and More on Morphisms, Lemma 37.19.3.
Our task is to extend this construction to the corresponding category of algebraic spaces X over S.
Let X be an algebraic space over S satisfying the equivalent conditions of Lemma 67.49.1. Let U \to X be a surjective étale morphism where U is a scheme. Set R = U \times _ X U with projections s, t : R \to U and j = (t, s) : R \to U \times _ S U so that X = U/R, see Spaces, Lemma 65.9.1. Observe that U and R are objects of \mathcal{C} by our assumptions on X and that the morphisms s and t are étale morphisms of schemes over S. By (a) we have the normalization morphisms \nu _ U : U^\nu \to U and \nu _ R : R^\nu \to R, by (d) we have morphisms s^\nu : R^\nu \to U^\nu , t^\nu : R^\nu \to U^\nu which define isomorphisms R^\nu \to R \times _{s, U} U^\nu and R^\nu \to U^\nu \times _{U, t} R. It follows that s^\nu and t^\nu are étale (as they are isomorphic to base changes of étale morphisms). The induced morphism j^\nu = (t^\nu , s^\nu ) : R^\nu \to U^\nu \times _ S U^\nu is a monomorphism as it is equal to the composition
\begin{align*} R^\nu & \to (U^\nu \times _{U, t} R) \times _ R (R \times _{s, U} U^\nu ) \\ & = U^\nu \times _{U, t} R \times _{s, U} U^\nu \\ & \xrightarrow {j} U^\nu \times _ U (U \times _ S U) \times _ U U^\nu \\ & = U^\nu \times _ S U^\nu \end{align*}
The first arrow is the diagonal morphism of \nu _ R. (This tells us that R^\nu is a subscheme of the restriction of R to U^\nu .) A formal computation with fibre products using property (d) shows that R^\nu \times _{s^\nu , U^\nu , t^\nu } R^\nu is the normalization of R \times _{s, U, t} R. Hence the étale morphism c : R \times _{s, U, t} R \to R extends uniquely to c^\nu by (d). The morphism c^\nu is compatible with the projection \text{pr}_{13} : U^\nu \times _ S U^\nu \times _ S U^\nu \to U^\nu \times _ S U^\nu . Similarly, there are morphisms i^\nu : R^\nu \to R^\nu compatible with the morphism U^\nu \times _ S U^\nu \to U^\nu \times _ S U^\nu which switches factors and there is a morphism e^\nu : U^\nu \to R^\nu compatible with the diagonal morphism U^\nu \to U^\nu \times _ S U^\nu . All in all it follows that j^\nu : R^\nu \to U^\nu \times _ S U^\nu is an étale equivalence relation. At this point we may and do set X^\nu = U^\nu /R^\nu (Spaces, Theorem 65.10.5). Then we see that we have U^\nu = X^\nu \times _ X U by Groupoids, Lemma 39.20.7.
What have we shown in the previous paragraph is this: for every algebraic space X over S satisfying the equivalent conditions of Lemma 67.49.1 if we choose a surjective étale morphism g : U \to X where U is a scheme, then we obtain a cartesian diagram
\xymatrix{ X^\nu \ar[d]_{\nu _ X} & U^\nu \ar[l]^{g^\nu } \ar[d]^{\nu _ U} \\ X & U \ar[l]_ g }
of algebraic spaces. This immediately implies that X^\nu is a normal algebraic space and that \nu _ X is a integral morphism. This gives part (1) of the lemma.
We will show below that the morphism \nu _ X : X^\nu \to X up to unique isomorphism is independent of the choice of g, but for now, if X is a scheme, we choose \text{id} : X \to X so that it is clear that we have part (2) of the lemma.
We still have to prove parts (3) and (4). Let g : U \to X and \nu _ X : X^\nu \to X and g^\nu : U^\nu \to X^\nu be as above. Let Z be a normal scheme and let h : Z \to U and a : Z \to X^\nu be morphisms over S such that g \circ h = \nu _ X \circ a and such that every irreducible component of Z dominates an irreducible component of U (via h). By Morphisms, Lemma 29.54.5 part (4) we obtain a unique morphism h^\nu : Z \to U^\nu such that h = \nu _ U \circ h^\nu . Picture:
\xymatrix{ X^\nu \ar[d]_{\nu _ X} & U^\nu \ar[l]^{g^\nu } \ar[d]^{\nu _ U} & Z \ar[l]^{h^\nu } \ar@/_1em/[ll]_ a \ar[dl]^ h \\ X & U \ar[l]_ g }
Observe that a = g^\nu \circ h^\nu . Namely, since the square with corners X^\nu , X, U^\nu , U is cartesian, this follows immediately from the fact that h^\nu is unique (given h). In other words, given h : Z \to U as above (and not a) there is a unique morphism a : Z \to X^\nu with \nu _ X \circ a = g \circ h.
Let f : X \to Y be as in part (3) of the statement of the lemma. Suppose we have chosen surjective étale morphisms U \to X and V \to Y where U and V are schemes such that f lifts to a morphism g : U \to V. Then g \in \text{Arrows}(\mathcal{C}) and we obtain a unique morphism g^\nu : U^\nu \to V^\nu compatible with \nu _ U and \nu _ V. However, then the two morphisms
R^\nu = U^\nu \times _{X^\nu } U^\nu \to U^\nu \to V^\nu \to Y^\nu
must be the same by our comments in the previous paragraph (applied with Y in stead of X). Since X^\nu is constructed by taking the quotient of U^\nu by R^\nu it follows that we obtain a (unique) morphism f^\nu : X^\nu \to Y^\nu as stated in (3).
To see that the construction of X^\nu is independent of the choice of g : U \to X surjective étale, apply the construction in the previous paragraph to \text{id} : X \to X and a morphism U' \to U between étale coverings of X. This is enough because given any two étale coverings of X there is a third one which dominates both. The reader shows that the morphism between the two normalizations constructed using either U' \to X or U \to X becomes an isomorphism after base change to U' and hence was an isomorphism. We omit the details.
We omit the proof of (4) which is similar; hint use part (d) above.
\square
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