In order to define a kernel in the general category-theoretical sense, C needs to have zero morphisms.
In concrete categories, one can thus take a subset of K′ for K, in which case, the morphism k is the inclusion map.
That is, the kernel of a morphism is its cokernel in the opposite category, and vice versa.
Conversely, in a preadditive category, every binary equaliser can be constructed as a kernel.
To be specific, the equaliser of the morphisms f and g is the kernel of the difference g − f. In symbols: It is because of this fact that binary equalisers are called "difference kernels", even in non-preadditive categories where morphisms cannot be subtracted.
In this situation, the kernel of the cokernel of any morphism (which always exists in an abelian category) turns out to be the image of that morphism; in symbols: When m is a monomorphism, it must be its own image; thus, not only are abelian categories normal, so that every monomorphism is a kernel, but we also know which morphism the monomorphism is a kernel of, to wit, its cokernel.