= Pseudo-epimorphism factorization through the kernel of a cokernel
{title2=$h=ie,\quad i=\ker(\operatorname{coker}h)$}
In a <pointed category> with <categorical kernels> and <categorical cokernels>, where every <monomorphism> is normal, every $h$ factors through $i=\ker(\operatorname{coker}h)$ as $h=ie$, with $e$ a <pseudo-epimorphism>. If $te=0$, write $k=\ker t$. Then $e$ factors through $k$, and the monic $ik$ is a <categorical kernel> of some $q$ by normality. Since $qh=0$, the cokernel of $h$ forces $qi=0$. Hence $i$ factors through $ik$, forcing $k$ to be an <isomorphism> and therefore $t=0$. This establishes the factorization without assuming the stronger <abelian category> axioms.
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