Epimorphisms of schemes

Question

Assume we have a morphism of schemes between two affine schemes. Is it true that it is an epimorphism in the category of schemes iff it is an epimorphism in the full subcategory of affine schemes?

Answer 1

No. Consider the two different inclusions of the affine line in the affine line with doubled origin, precomposed by the (very affine) inclusion of the affine line without the origin in the affine line.

Let $A$ be a commutative ring. The inclusion $A[X] → A[X]_X$ is mono in the category of commutative rings, so for $U = \operatorname{Spec} A[X]_X$, the inclusion $ι \colon U → \mathbb A^1_A$ is epi in the category of affine schemes, but for the affine line over $A$ with doubled origin $Z_A$, the two different inclusions $j \colon \mathbb A^1_A → Z_A$ and $k \colon \mathbb A^1_A → Z_A$ give both the same composition $jι = kι$, so $ι$ is not epi in the category of schemes.