Diagonal morphism

Let $f:Y\to X$ be a morphism of schemes. The two identity maps from $Y$ to itself have the same composite $f$ to $X$. The universal property of the fiber product therefore gives a unique morphism

whose composites with both projections $Y\times_XY\to Y$ are the identity. This is the diagonal morphism of $f$.

The notation is the scheme-theoretic analogue of the point-set diagonal $y\mapsto(y,y)$. Its scheme structure also records infinitesimal information about the fibers of $f$, which is why properties of the diagonal characterize separation and ramification conditions.