# Monomorphism & epimorphism in the category of schemes

Is there a morphism in the category of schemes which is simultaneously a monomorphism and an epimorphism yet is not an isomorphism? "Nicer" examples are preferred (e.g. with integral Noetherian separated source and target).

There are such morphisms in the category of commutative unital rings e.g. $$\mathbb{Z}\rightarrow \mathbb{Q}$$ (https://math.stackexchange.com/a/2792241/690882, https://mathoverflow.net/q/109). An epimorphism in the category of commutative unital rings defines a monomorphism in the category of schemes (I think) but a monomorphism in the category of commutative unital rings is not necessarily an epimorphism in the category of schemes (https://math.stackexchange.com/a/3302479/690882).

Sure, take a local domain $$A$$ and its field of fractions $$Q = \operatorname{Quot} A$$. Then $$\operatorname{Spec} Q → \operatorname{Spec} A$$ is epi and mono in the category of schemes, but no isomorphism.