Diagonal morphism (${\Delta }_{X/S}:X\to X{\times }_{S}X$)

The diagonal morphism sends a point to the pair consisting of that point twice. Every diagonal of schemes is a locally closed immersion, and a morphism is separated exactly when its diagonal is a closed immersion.

For morphisms $f,g:X\to Y$ over a base $S$, their scheme-theoretic coincidence locus is the fibre productIt is the equalizer of $f$ and $g$, hence is universal among subschemes on which they agree. It is locally closed because every diagonal is an immersion, and closed when $Y\to S$ is separated.