The following question came up when reading Hartshorne's proof of the valuative criterion.
Let $f\colon X\to Y$ be a $Y$-scheme and let $K$ be a field. Let $g\colon \operatorname{Spec}K\to X \times_Y X$ be a morphism with image in the diagonal $\Delta(X)$. Why is it true that $g$ factors through the diagonal morphism?
Is there a more general result giving a sufficient, purely topological (or even only set-theoretic) condition for a morphism to factor through the diagonal morphism?
Any kind of help and information is appreciated.