Nice fact: Suppose f:X->Y is a map of schemes and Z⊆Y is a subscheme (locally closed immersion) containing the set-image of X. If X and Z are reduced, then it follows that f factors through Z. This is nice because it makes "factoring through" purely a consideration of the underlying topological spaces.
So now I'm wondering, to what extent does "reduced" allow us to think only terms of topological spaces? Suppose we weaken the assumption that Z→Y is an inclusion. When can we say f factors through Z? More precisely:
Suppose X,Z are reduced schemes, f:X→Y and g:Z→Y are scheme morphisms such that f factors through g in Top. When does f factor through g in Sch?
I know the answer is "not always", for example if Y is a field and X,Z are incomparable field extensions of Y (in Ringop). But does anyone know any positive results we can state here?