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Take any morphism $h:A\to B$ and let $c:B\to C$ be its categorical cokernel. Let $i:I\to B$ be the categorical kernel of $c$. Since $ch=0$, kernel universality gives $e:A\to I$ with $h=ie$. The map $i$ is a monomorphism. We must establish the pseudo-epimorphism property of $e$, rather than merely name $I$ an image.

Suppose $t:I\to Z$ satisfies $te=0$. Let $k:K\to I$ be its categorical kernel. Then $e=ka$ for some $a:A\to K$. The composite $ik$ is a monomorphism, so the hypothesis that every monomorphism is normal provides a map $q:B\to D$ of which $ik$ is a categorical kernel, up to its unique compatible isomorphism. Now $qh=qika=0$. The categorical cokernel property of $c$ gives $q=\bar{q}c$, and therefore $qi=\bar{q}ci=0$.

Since $ik$ is the categorical kernel of $q$, the map $i$ factors through it: $i=ik\ell$ for some $\ell :I\to K$. Cancel the monomorphism $i$ to get $k\ell =1_I$. As $k$ is also a monomorphism, $\ell k=1_K$, so $k$ is an isomorphism. Finally $t=tk\ell =0$, proving that $e$ is a pseudo-epimorphism.

ThusThe pseudo-epimorphism factorization through the kernel of a cokernel proof uses a pointed category, existence of categorical kernels and categorical cokernels, and normality of every monic map. It does not silently replace pseudo-epic cancellation against zero by the stronger epimorphism condition.

In a pointed category with categorical kernels and categorical cokernels, where every monomorphism is normal, every $h$ factors through $i=\ker (\mathrm{coker}h)$ 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.