= Closure of normal monomorphisms forces abelianness
In an <additive category> with finite <categorical limits> and <colimits> in which every <epimorphism> is a <conormal epimorphism>, closure of <normal monomorphisms> under composition forces every <monomorphism> to be normal. To prove this, write a monomorphism $m$ as $kg$, where $k=\ker\operatorname{coker}m$, and put $l=\ker\operatorname{coker}g$. If $kl=\ker t$ were normal, $tm=0$ would force $tk=0$, hence $l$ would be an <isomorphism>. This makes $\operatorname{coker}g=0$, so $g$ is epic as well as monic. An epic monomorphism which is a cokernel is invertible, forcing $m$ to be normal. Contrapositively, any failure of normality provides two normal monomorphisms whose composite is not normal.
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