TITLE: Identify the universal property of kernels
QUESTION [8 upvotes]: I'm reading Mac Lane's "Categories for the Working Mathematician". I found the following sentence in page 59 of it:

Similarly, the kernel of a homomorphism (in $\mathbf{Ab}$, $\mathbf{Grp}$, $\mathbf{Rng}$, $R$-$\mathbf{Mod}$, . . .) is a universal, more exactly, a universal for a suitable contravariant functor.

I know how to express a quotient by a universal property in a category. But I really have trouble understanding how to express the kernel of a homomorphism as having a universal property.

REPLY [2 votes]: In an abelian category, a kernel of a morphism $f\colon X\to Y$ is a solution of the following universal problem: find a morphism $i\colon K\to X$ such that:

$f\mkern 1.5mu i=0$.
for any morphism $u\colon Z\to X$ such that $f\mkern 1.5mu u=0$, there exists a morphism $v:Z\to K$ such that $u=i\mkern 1.5mu v$.

More generally, in a category with zero morphisms, $\ker f$ can be defined as the equaliser of $f$ and $\, 0_{XY}$ (again a universal problem).