This has H = 2; II.3.13d can be used to solve it.
Let be the open set where the two morphisms agree; let be the composition.
- From the assumption construct a map ; let be its restriction to
- Argue (perhaps by a universal property) that , in particular there is a factorization
- use that X is reduced and II.3.11d to show in fact
- apply either projection to get a morphism ; it follows
- play around with stuff like to show
- One example to show all assumptions are necessary: . Consider the maps given by the ring map in the other direction: and , find an open set where they agree. For an exmaple with not separated let it be the affine line with the doubled origin and affine line.