The kernel pair of a morphism $f:X\to Y$ in a category $C$ is a pair of morphisms $R\,\rightrightarrows \, X$ which form a limit of the diagram

$\array{
X & & & & X \\
& \searrow^f & & \swarrow_f \\
& & Y \\
}$

We can think of this as the fiber product$X \times_Y X$ of $X$ with itself over $Y$, or as the pullback of $f$ along itself.

Properties

The kernel pair is always a congruence on $X$; informally, $R$ is the subobject of $X \times X$ consisting of pairs of elements which have the same value under $f$ (sometimes called the ‘kernel’ of a function in $\Set$).

The coequalizer of the kernel pair, if it exists, is supposed to be the “object of equivalence classes” of the internal equivalence relation$R$. In other words, it is the quotient object of $X$ in which generalized elements are identified if they are mapped by $f$ to equal values in $Y$. In a regular category (at least), this can be identified with a subobject of $Y$ called the image of $f$.

If a morphism has a kernel pair and is a coequalizer, then it is the coequalizer of its kernel pair. This is a special case of the correspondence of generalized kernels in enriched categories.