Exactness of kernel and cokernel sequences under endpoint hypotheses

Assume the top row is exact and $k$ is monic. Choose kernels $h:\ker (\alpha )\to X$, $i:\ker (\beta )\to Y$, and $j:\ker (\gamma )\to Z$. By [L3], the commutative diagram induces morphisms $a:\ker (\alpha )\to \ker (\beta ),b:\ker (\beta )\to \ker (\gamma )$ with $ia=fh,jb=gi\mathrm{.}$

To prove exactness of $\ker (\alpha )\stackrel{a}{\to }\ker (\beta )\stackrel{b}{\to }\ker (\gamma ),$ apply the covering criterion [L1] to the pair $(a,b)$. Let $u:T\to \ker (\beta )$ satisfy $bu=0$. Then $giu=jbu=0,$ so exactness of the top row gives an object $P$, an epimorphism $e:P\to T$, and a morphism $x:P\to X$ with $iue=fx\mathrm{.}$

Applying $\beta$ to the displayed equality gives $0=\beta iue=k\alpha x\mathrm{.}$ Since $k$ is monic, $\alpha x=0$. The kernel property of $h$ therefore gives $x^{\mathrm{\prime }}:P\to \ker (\alpha )$ with $h{x}^{\mathrm{\prime }}=x$. Now $iue=fh{x}^{\mathrm{\prime }}=ia{x}^{\mathrm{\prime }},$ so monicity of $i$ from [L4] yields $ue=a{x}^{\mathrm{\prime }}$. Also $jba=gia=gfh=0,$ so monicity of $j$ from [L4] gives $ba=0$. Thus the pair $(a,b)$ satisfies both parts of the covering criterion [L1], and the kernel sequence is exact.

The cokernel statement is the formal dual of steps 1.1 to 3.1 in the opposite abelian category: bottom-row exactness becomes top-row exactness, the epicity of $g$ becomes monicity of $g^{\mathrm{o}\mathrm{p}}$, kernels become cokernels, and [L2] transports the resulting exact sequence back to the original category.

Therefore both displayed induced sequences are exact under the stated endpoint hypotheses.