Equalizer

Summary

The equalizer of two parallel morphisms $f, g \colon X \to Y$ in a category is a universal morphism $e \colon E \to X$ with $f \circ e = g \circ e$. It is the limit of the diagram formed by $f$ and $g$. In sets it is the subset where $f$ and $g$ agree, and in abelian groups it is the kernel of $f - g$. Every equalizer is a monomorphism, and products together with equalizers build all limits.