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 Ring<sup>op</sup>).  But does anyone know any positive results we can state here?