The kernel row and cokernel row of a morphism of short exact sequences are exact at two nodes each

Because both rows are short exact, [L1] says that the left square is a morphism between exact pairs and that $i^{\mathrm{\prime }}$ is monic. Therefore [L2] applies and gives exactness of the induced kernel sequence $\ker (a)\to \ker (b)\to \ker (c)$ at $\ker (b)$.

Let $u:\ker (a)\to \ker (b)$ be the induced map and choose kernel arrows $k_a:\ker (a)\to A$ and $k_b:\ker (b)\to B$. If $us=ut$, then $i{k}_{a}s=k_{b}us=k_{b}ut=i{k}_{a}t\mathrm{.}$ The map $i$ is monic by [L1], and $k_a$ is monic by [L3], so $s=t$. Hence $u$ is monic, which is exactly exactness of $0\to \ker (a)\to \ker (b)$ at $\ker (a)$.

Passing to the opposite category turns the diagram into a morphism of short exact sequences again. Applying steps 1.1 and 1.2 there and transporting the result back with [L4] yields exactness of the induced cokernel sequence at $\mathrm{coker}(b)$ and at $\mathrm{coker}(c)$.

Hence the kernel row is exact at $\ker (a)$ and $\ker (b)$, and the cokernel row is exact at $\mathrm{coker}(b)$ and $\mathrm{coker}(c)$.