Let $A$ be an additive category with kernels and cokernels. A morphism $f$ is called strict if the natural morphism from the coimage to the image is an isomorphism.
In *Schneiders: Quasi-abelian categories and sheaves* one finds a proof that the composition of strict morphisms is strict, so the strict morphisms form a subcategory, which we denote by $A\_{\mathrm{st}}$.


Is $A\_{\mathrm{st}}$ abelian? Where can I find a proof or counterexample?



Is the subcategory of strict morphisms abelian?
I do not believe that the composition of strict morphisms is strict, proposition 1.1.7 of Schneider's *Quasi-Abelian Categories and Sheaves* claims this only for strict epimorphisms as well as for strict monomorphisms.


I think that rather the opposite of the OP's claim is true:



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> If all compostions of strict morphisms in an additive category with kernels and cokernels are again strict, then the category is abelian.
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To prove this we write an arbitrary morphism $f:X\to Y$ as a composition of a strict epimorphism and a strict monomorphism $$f=\pi\_Y\circ [id\_X,f]: X\to X\times Y\to Y.$$


