Snake Lemma

Consider

where

are the pushouts

Then the zero morphisms induce a morphism

where

is a cokernel (cf. cokernel, pushout and exact esquence)

Then it follows, as is a monomorphism (cf. monomorphism mirrored by pushouts in a regular category, abelian category as regular category) that

(cf. monomorphism as kernel of the cokernel)

by a dual argument we have that

is a kernel

(cf. kernel, pullback and exact sequence)

Then by commutativity of

there exists an induced morphism to

It follows that the dashed morphism commutes with , as

and uniqueness of the universal property

Hence is the zero morphism, as is the zero morphism.

Then using the universal property of as cokernel (here is a kernel) results in the boundary morphism