of the kernel of a morphism. In concrete categories, one can thus take a subset of X 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.

Not every morphism needs to have a kernel, but if it does, then all its kernels