As for every universal property, there is a unique isomorphism between two kernels of the same morphism, and the morphism k is always a monomorphism (in the categorical sense). So, it is common to talk of the kernel of a morphism. In concrete categories, one can thus take a subset of ' for K, in which case, the morphism k is the inclusion map. This allows one to talk of K as the kernel, since k is implicitly defined by K. There are non-concrete categories, where one can similarly define a "natural" kernel, such that K defines k implicitly.
As for every universal property there is a unique isomorphism between two kernels of the same morphism and the morphism k is always a monomorphism (in the categorical sense) So it is common to talk of the kernel of a morphism In concrete categories one can thus take a subset of ' for K in which case the morphism k is the inclusion map This allows one to talk of K as the kernel since k is implicitly defined by K There are non-concrete categories where one can similarly define a "natural" kernel such that K defines k implicitly