 Kernel squares in an abelian category

In a commutative diagram of two kernel rows, a monic arrow between the final objects makes the square between their kernels and middle objects a pullback in a category. The kernel universal property factors any categorical cone uniquely. Conversely, if the square between middle and final objects is a pullback, the induced arrow between their kernels in a category is an isomorphism, since both kernels have the same universal property after pullback.