Proposition

Let $X$ be a scheme (resp. a locally noetherian scheme), $f: X\to Y$ a morphism of schemes (resp. a morphism locally of finite type). The following conditions are equivalent.

1. $f$ is separated.

2. The diagonal morphism $X\to X\times_Y X$ is quasicompact, and for every affine scheme $Y' = Spec A$ in which $A$ is a valuation ring (resp. a discrete valuation ring), any two morphisms from $Y'\to X$ which coincide at the generic point of $Y'$ are equal.

3. The diagonal morphism $X\to X\times_Y X$ is quasicompact, and for every affine scheme of the form $Y' = Spec A$ in which $A$ is a valuation ring (resp. a discrete valuation ring), any two sections of $X' = X(Y')$ which coincide at the generic point of $Y'$ are equal.