A question related to Stein factorisation

Question

I am trying to prove this theorem, Let $X=\prod_{i=1}^n X_i $.

Theorem: Let $f: X \rightarrow X_i$ is a projection map (i.e., surjective with connected fibers) and $ g: X\rightarrow Z$ is a proper morphism that is constant on the fibers of $f$, then there exist a unique morphism $h: X_i \rightarrow Z$ such that $g=h \cdot f$.

I am stuck proving this. Is it a standard theorem, that I can locate in Hartshorne exercise? Or is it wrong?

I know there is something called $Stein$ $factorisation$,(https://en.wikipedia.org/wiki/Stein_factorization#CITEREFEGA) that says,

For a proper morphism $F: X \rightarrow Y$, one can write $F=G\cdot F'$, where $G: Y' \rightarrow Y$ a finite morphism and $F': X \rightarrow Y'$ a proper morphism with direct image of structure sheaf is a structure sheaf.

Now, I am stuck on how to apply and prove our case, if possible.

Any help regarding this would be immensely helpful. Thank you.

Answer 1

By faithfully flat descent for morphisms of schemes, if $f \colon X \to X_i$ is faithfully flat and $g\colon X\to Z$ is an arbitrary morphism of schemes, then there exists a unique morphism $h \colon Z_i \to Z $ such that $g = h\cdot f $ if (and in fact only if) the pullbacks of $h$ to $X \times_{X_i} X$ along the two projections $X \times_{X_i} X \to X$ are equal.

Your conditions "surjective with connected fibers" do not imply faithfully flat, but the projection $f$ is faithfully flat: Any morphism from a nonempty scheme to the spectrum of a field is faithfully flat so in particular $\prod_{j \neq i }X_j \to \operatorname{Spec} k$ is, and faithfully flat morphisms are preserved by base change, so $f$ is as well.

Your condition "constant on the fibers of $f$" implies that the pullbacks of $h$ to $X \times_{X_i} X$ along the two projections $X \times_{X_i} X \to X$ are equal as long as $X \times_{X_i} X$ is reduced (for example if the $X_i$ are all reduced over a field of characteristic zero): In this case, it suffices to check the equality at every point of $X \times_{X_i} X$, which lies in the product of one of the fibers of $f$ with itself, so it suffices to check that the value of $h$ at the two points in the fiber are the same, which is the same as constancy.

Being a little careful I think one can check the implication just using the assumption that $X_i$ is reduced. Your desired statement is certainly not true if $X_i$ is not reduced.